Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.14 and add it to the current value.
Final: {"answer": [-1.258, 0.351, 1.904], "unit": "temperature_units"}

Answer: -1.258

Answer: 0.351

Answer: 1.904

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-0.9, right=2.4; current interior values are [-1.65, 0.4, 2.1]. Use r=0.14. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.14 and add it to the current value.


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 108.5 + 273.15.
Final: {"answer": 381.65, "unit": "K"}

Answer: 381.65

Prompt: Convert 108.5 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute 108.5 + 273.15.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=234 degrees and keep the unit as newtons.
Final: {"answer": [-291.2476, -400.8679], "unit": "N"}

Answer: -291.2476

Answer: -400.8679

Prompt: A vector has magnitude 495.5 N at 234 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=234 degrees and keep the unit as newtons.

[magnitude * math.cos(math.radians(angle_deg)), magnitude * math.sin(math.radians(angle_deg))]


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.45, -22.5], "unit": "dimensionless"}

Answer: 8.45

Answer: -22.5

Prompt: A line passes through points (5.75, 26.0875) and (10, 62). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. lever arm times force has torque dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In model card equation case 491384, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
lever arm times force has torque dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def running_total_005110(values):
    total = 0
    out = []
    for value in values:
        total += value
        out.append(total)
    return out
```
Final: {"answer": "python_code", "function": "running_total_005110"}

Answer: python_code

Prompt: Write a Python function running_total_005110(values) that returns cumulative sums as a new list. It must not mutate the input list. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 1.7 kg and a = 14.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 24.31, "unit": "N"}

Answer: 24.31

Prompt: A cart has mass 1.7 kg and measured acceleration 14.3 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 1.7 kg and a = 14.3 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 39.05^2 and multiply by 0.5 * 28.4.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 21653.6155, "unit": "J"}

Answer: 21653.6155

Prompt: A body of mass 28.4 kg moves at 39.05 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 39.05^2 and multiply by 0.5 * 28.4.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1670 * 0.565 * 106.
Final: {"answer": 100016.3, "unit": "J"}

Answer: 100016.3

Prompt: A 1670 g sample has specific heat 0.565 J/(g*K). How much heat is needed to raise its temperature by 106 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1670 * 0.565 * 106.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=218.4 V and R=2.6 ohm.
Final: {"answer": 18345.6, "unit": "W"}

Answer: 18345.6

Prompt: A resistor has resistance 2.6 ohm and voltage 218.4 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=218.4 V and R=2.6 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=496 mg, t=121 h, and T_half=107.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 227.3254, "unit": "mg"}

Answer: 227.3254

Prompt: A radioactive sample starts with 496 mg. Its half-life is 107.5 hours. How many mg remain after 121 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=496 mg, t=121 h, and T_half=107.5 h.
The remaining amount has the same mass unit as the initial amount.

initial_amount * 0.5 ** (elapsed_time / half_life)


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.11 y, so each step multiplies y by 1 - h*0.11.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": 2.4327, "unit": "state_units"}

Answer: 2.4327

Prompt: Use explicit Euler on dy/dt = -0.11 y with y(0)=5.5, step size h=1, for 7 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.11 y, so each step multiplies y by 1 - h*0.11.
Apply that update for 7 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-2.7)=-11.65 and f(1.7)=-13.85.
3. The interval width is 4.4.
Final: {"answer": -56.1, "unit": "area_units"}

Answer: -56.1

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-0.5x+-13 from x=-2.7 to x=1.7. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-2.7)=-11.65 and f(1.7)=-13.85.
The interval width is 4.4.

(x1 - x0) * ((slope * x0 + intercept) + (slope * x1 + intercept)) / 2


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.02 and add it to the current value.
Final: {"answer": [-1.762, 2.192, -1.172], "unit": "temperature_units"}

Answer: -1.762

Answer: 2.192

Answer: -1.172

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=0.75, right=1.45; current interior values are [-1.9, 2.35, -1.3]. Use r=0.02. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.02 and add it to the current value.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=210 degrees and keep the unit as newtons.
Final: {"answer": [-177.9682, -102.75], "unit": "N"}

Answer: -177.9682

Answer: -102.75

Prompt: A vector has magnitude 205.5 N at 210 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=210 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity divided by time is acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In simulation spec case 526151, check dimensional consistency for x = v / t. The left side has dimensions m and the right side has dimensions m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity divided by time is acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def finite_differences_005128(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_005128"}

Answer: python_code

Prompt: Write a Python function finite_differences_005128(values) that returns the list of successive differences values[i+1] - values[i]. It must return [] for fewer than two values. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

assert finite_differences_005128([5]) == []

assert finite_differences_005128([]) == []


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: recurrence, objective, tie_breaking_rule.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["recurrence", "objective", "tie_breaking_rule"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a segmentation scoring problem with 3615 items and budget 43515, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: recurrence, objective, tie_breaking_rule.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 15.1 kg and a = 11.45 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 172.895, "unit": "N"}

Answer: 172.895

Prompt: A cart has mass 15.1 kg and measured acceleration 11.45 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 15.1 kg and a = 11.45 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 13.05^2 and multiply by 0.5 * 18.1.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1541.2376, "unit": "J"}

Answer: 1541.2376

Prompt: A body of mass 18.1 kg moves at 13.05 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 13.05^2 and multiply by 0.5 * 18.1.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=42 degrees through the sine of 2 theta.
3. Substitute v=84.9 m/s and g=9.81 m/s^2.
Final: {"answer": 730.7364, "unit": "m"}

Answer: 730.7364

Prompt: A projectile is launched on level ground with no air resistance at speed 84.9 m/s and angle 42 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=42 degrees through the sine of 2 theta.
Substitute v=84.9 m/s and g=9.81 m/s^2.

speed ** 2 * math.sin(math.radians(2 * angle_deg)) / gravity


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.6 * 8.314 * 552 / 2.9.
Final: {"answer": 7279.6237, "unit": "kPa"}

Answer: 7279.6237

Prompt: An ideal gas sample has n=4.6 mol, T=552 K, and V=2.9 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.6 * 8.314 * 552 / 2.9.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1305 * 3.585 * 46.
Final: {"answer": 215207.55, "unit": "J"}

Answer: 215207.55

Prompt: A 1305 g sample has specific heat 3.585 J/(g*K). How much heat is needed to raise its temperature by 46 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1305 * 3.585 * 46.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=104.9 V and R=122.4 ohm.
Final: {"answer": 89.902, "unit": "W"}

Answer: 89.902

Prompt: A resistor has resistance 122.4 ohm and voltage 104.9 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=104.9 V and R=122.4 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=22 mg, t=218.5 h, and T_half=15.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0013, "unit": "mg"}

Answer: 0.0013

Prompt: A radioactive sample starts with 22 mg. Its half-life is 15.5 hours. How many mg remain after 218.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=22 mg, t=218.5 h, and T_half=15.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 6.05 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 605.5263, "unit": "g"}

Answer: 605.5263

Prompt: A sample contains 6.05 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.05 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4985 mL to 4.985 L.
3. Compute M = 1.335 / 4.985.
Final: {"answer": 0.2678, "unit": "mol/L"}

Answer: 0.2678

Prompt: A solution contains 1.335 mol of solute in 4985 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4985 mL to 4.985 L.
Compute M = 1.335 / 4.985.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.38 y, so each step multiplies y by 1 - h*0.38.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 0.2074, "unit": "state_units"}

Answer: 0.2074

Prompt: Use explicit Euler on dy/dt = -0.38 y with y(0)=36.5, step size h=1.15, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.38 y, so each step multiplies y by 1 - h*0.38.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(9.8)=-44.56 and f(14.6)=-62.32.
3. The interval width is 4.8.
Final: {"answer": -256.512, "unit": "area_units"}

Answer: -256.512

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.7x+-8.3 from x=9.8 to x=14.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(9.8)=-44.56 and f(14.6)=-62.32.
The interval width is 4.8.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.22 and add it to the current value.
Final: {"answer": [-0.685, 0.551, 1.158], "unit": "temperature_units"}

Answer: -0.685

Answer: 0.551

Answer: 1.158

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-0.75, right=1.9; current interior values are [-1.4, 1.2, 0.85]. Use r=0.22. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.22 and add it to the current value.


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 83.5 + 273.15.
Final: {"answer": 356.65, "unit": "K"}

Answer: 356.65

Prompt: Convert 83.5 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute 83.5 + 273.15.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [3.1, 9.75], "unit": "dimensionless"}

Answer: 3.1

Answer: 9.75

Prompt: A line passes through points (-6.5, -10.4) and (-0.25, 8.975). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. grams cancel with J/(g*K), and kelvin cancels, leaving joules
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In quality-control check case 373948, check dimensional consistency for Q = m * c * DeltaT. The left side has dimensions J and the right side has dimensions J. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
grams cancel with J/(g*K), and kelvin cancels, leaving joules
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def center_values_005146(values):
    if not values:
        raise ValueError('center_values requires at least one value')
    mean = sum(values) / len(values)
    return [value - mean for value in values]
```
Final: {"answer": "python_code", "function": "center_values_005146"}

Answer: python_code

Prompt: Write a Python function center_values_005146(values) that subtracts the arithmetic mean from each value and returns a new list. Raise ValueError for an empty input. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    center_values_005146([])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: rate_constant, number_of_steps.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["rate_constant", "number_of_steps"]}

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=71.5 and h=0.5, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: rate_constant, number_of_steps.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 19.2 kg and a = 10.5 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 201.6, "unit": "N"}

Answer: 201.6

Prompt: A cart has mass 19.2 kg and measured acceleration 10.5 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 19.2 kg and a = 10.5 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 1.6 * 8.314 * 360 / 106.0.
Final: {"answer": 45.178, "unit": "kPa"}

Answer: 45.178

Prompt: An ideal gas sample has n=1.6 mol, T=360 K, and V=106.0 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 1.6 * 8.314 * 360 / 106.0.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1095 * 2.245 * 78.
Final: {"answer": 191745.45, "unit": "J"}

Answer: 191745.45

Prompt: A 1095 g sample has specific heat 2.245 J/(g*K). How much heat is needed to raise its temperature by 78 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1095 * 2.245 * 78.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=143.2 V and R=92.9 ohm.
Final: {"answer": 220.7346, "unit": "W"}

Answer: 220.7346

Prompt: A resistor has resistance 92.9 ohm and voltage 143.2 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=143.2 V and R=92.9 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=879 mg, t=128.5 h, and T_half=103.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 371.746, "unit": "mg"}

Answer: 371.746

Prompt: A radioactive sample starts with 879 mg. Its half-life is 103.5 hours. How many mg remain after 128.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=879 mg, t=128.5 h, and T_half=103.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.82 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 508.0399, "unit": "g"}

Answer: 508.0399

Prompt: A sample contains 2.82 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 2.82 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4560 mL to 4.56 L.
3. Compute M = 2.495 / 4.56.
Final: {"answer": 0.5471, "unit": "mol/L"}

Answer: 0.5471

Prompt: A solution contains 2.495 mol of solute in 4560 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4560 mL to 4.56 L.
Compute M = 2.495 / 4.56.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 27968.0 / 1000.
Final: {"answer": 27.968, "unit": "kg"}

Answer: 27.968

Prompt: Convert 27968 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 27968.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=293 degrees and keep the unit as newtons.
Final: {"answer": [165.8654, -390.7543], "unit": "N"}

Answer: 165.8654

Answer: -390.7543

Prompt: A vector has magnitude 424.5 N at 293 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=293 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [1.4, 12.75], "unit": "dimensionless"}

Answer: 1.4

Answer: 12.75

Prompt: A line passes through points (-6, 4.35) and (3.5, 17.65). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. lever arm times force has torque dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In quality-control check case 44857, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
lever arm times force has torque dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def threshold_flags_005164(values, threshold):
    return [value >= threshold for value in values]
```
Final: {"answer": "python_code", "function": "threshold_flags_005164"}

Answer: python_code

Prompt: Write a Python function threshold_flags_005164(values, threshold) that returns a list of booleans showing whether each value is greater than or equal to threshold. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

assert threshold_flags_005164([1, 2, 3], 2) == [False, True, True]

assert threshold_flags_005164([], 5) == []

assert threshold_flags_005164([-1, 0], 0) == [False, True]


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: rate_constant, number_of_steps.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["rate_constant", "number_of_steps"]}

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=8.5 and h=1.5, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: rate_constant, number_of_steps.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 8.05 kg and a = 7.85 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 63.1925, "unit": "N"}

Answer: 63.1925

Prompt: A cart has mass 8.05 kg and measured acceleration 7.85 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 8.05 kg and a = 7.85 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=27 degrees through the sine of 2 theta.
3. Substitute v=31.9 m/s and g=9.81 m/s^2.
Final: {"answer": 83.9209, "unit": "m"}

Answer: 83.9209

Prompt: A projectile is launched on level ground with no air resistance at speed 31.9 m/s and angle 27 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=27 degrees through the sine of 2 theta.
Substitute v=31.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=124 V and R=190.2 ohm.
Final: {"answer": 80.8412, "unit": "W"}

Answer: 80.8412

Prompt: A resistor has resistance 190.2 ohm and voltage 124 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=124 V and R=190.2 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=262.5 mg, t=128.5 h, and T_half=80 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 86.2184, "unit": "mg"}

Answer: 86.2184

Prompt: A radioactive sample starts with 262.5 mg. Its half-life is 80 hours. How many mg remain after 128.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=262.5 mg, t=128.5 h, and T_half=80 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2985 mL to 2.985 L.
3. Compute M = 2.965 / 2.985.
Final: {"answer": 0.9933, "unit": "mol/L"}

Answer: 0.9933

Prompt: A solution contains 2.965 mol of solute in 2985 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2985 mL to 2.985 L.
Compute M = 2.965 / 2.985.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.39 y, so each step multiplies y by 1 - h*0.39.
3. Apply that update for 5 steps and round the final state.
Final: {"answer": 7.7836, "unit": "state_units"}

Answer: 7.7836

Prompt: Use explicit Euler on dy/dt = -0.39 y with y(0)=13, step size h=0.25, for 5 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.39 y, so each step multiplies y by 1 - h*0.39.
Apply that update for 5 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-2)=-22.5 and f(-0.1)=-7.87.
3. The interval width is 1.9.
Final: {"answer": -28.8515, "unit": "area_units"}

Answer: -28.8515

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=7.7x+-7.1 from x=-2 to x=-0.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-2)=-22.5 and f(-0.1)=-7.87.
The interval width is 1.9.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.28 and add it to the current value.
Final: {"answer": [-0.088, -0.57, 0.164], "unit": "temperature_units"}

Answer: -0.088

Answer: -0.57

Answer: 0.164

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.25, right=2.15; current interior values are [-0.9, -0.15, -0.9]. Use r=0.28. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.28 and add it to the current value.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 13587.0 / 1000.
Final: {"answer": 13.587, "unit": "kg"}

Answer: 13.587

Prompt: Convert 13587 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 13587.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=85 degrees and keep the unit as newtons.
Final: {"answer": [11.766, 134.4863], "unit": "N"}

Answer: 11.766

Answer: 134.4863

Prompt: A vector has magnitude 135 N at 85 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=85 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity divided by time is acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In student derivation case 383789, check dimensional consistency for x = v / t. The left side has dimensions m and the right side has dimensions m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity divided by time is acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def threshold_flags_005182(values, threshold):
    return [value >= threshold for value in values]
```
Final: {"answer": "python_code", "function": "threshold_flags_005182"}

Answer: python_code

Prompt: Write a Python function threshold_flags_005182(values, threshold) that returns a list of booleans showing whether each value is greater than or equal to threshold. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

assert threshold_flags_005182([1, 2, 3], 2) == [False, True, True]

assert threshold_flags_005182([], 5) == []

assert threshold_flags_005182([-1, 0], 0) == [False, True]


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 14.15 kg and a = 8 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 113.2, "unit": "N"}

Answer: 113.2

Prompt: A cart has mass 14.15 kg and measured acceleration 8 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 14.15 kg and a = 8 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 35.65^2 and multiply by 0.5 * 17.15.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 10898.1604, "unit": "J"}

Answer: 10898.1604

Prompt: A body of mass 17.15 kg moves at 35.65 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 35.65^2 and multiply by 0.5 * 17.15.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=67 degrees through the sine of 2 theta.
3. Substitute v=54.5 m/s and g=9.81 m/s^2.
Final: {"answer": 217.8001, "unit": "m"}

Answer: 217.8001

Prompt: A projectile is launched on level ground with no air resistance at speed 54.5 m/s and angle 67 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=67 degrees through the sine of 2 theta.
Substitute v=54.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 2.02 * 8.314 * 593 / 84.8.
Final: {"answer": 117.4411, "unit": "kPa"}

Answer: 117.4411

Prompt: An ideal gas sample has n=2.02 mol, T=593 K, and V=84.8 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2.02 * 8.314 * 593 / 84.8.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1100 * 2.565 * 92.
Final: {"answer": 259578.0, "unit": "J"}

Answer: 259578.0

Prompt: A 1100 g sample has specific heat 2.565 J/(g*K). How much heat is needed to raise its temperature by 92 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1100 * 2.565 * 92.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=169.7 V and R=34.6 ohm.
Final: {"answer": 832.3147, "unit": "W"}

Answer: 832.3147

Prompt: A resistor has resistance 34.6 ohm and voltage 169.7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=169.7 V and R=34.6 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=31.5 mg, t=254.5 h, and T_half=56 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 1.3497, "unit": "mg"}

Answer: 1.3497

Prompt: A radioactive sample starts with 31.5 mg. Its half-life is 56 hours. How many mg remain after 254.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=31.5 mg, t=254.5 h, and T_half=56 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.2 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 316.872, "unit": "g"}

Answer: 316.872

Prompt: A sample contains 7.2 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 7.2 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3145 mL to 3.145 L.
3. Compute M = 2.86 / 3.145.
Final: {"answer": 0.9094, "unit": "mol/L"}

Answer: 0.9094

Prompt: A solution contains 2.86 mol of solute in 3145 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3145 mL to 3.145 L.
Compute M = 2.86 / 3.145.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.12 and add it to the current value.
Final: {"answer": [-0.69, -0.282, -0.616], "unit": "temperature_units"}

Answer: -0.69

Answer: -0.282

Answer: -0.616

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.85, right=2.15; current interior values are [-1.2, 0.0, -1.15]. Use r=0.12. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.12 and add it to the current value.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 4929.5 / 100.
Final: {"answer": 49.295, "unit": "m"}

Answer: 49.295

Prompt: Convert 4929.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 4929.5 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=213 degrees and keep the unit as newtons.
Final: {"answer": [-261.6652, -169.9274], "unit": "N"}

Answer: -261.6652

Answer: -169.9274

Prompt: A vector has magnitude 312 N at 213 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=213 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [2.9, 23.25], "unit": "dimensionless"}

Answer: 2.9

Answer: 23.25

Prompt: A line passes through points (-17, -26.05) and (-16.5, -24.6). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 19.8^2 and multiply by 0.5 * 2.8.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 548.856, "unit": "J"}

Answer: 548.856

Prompt: A body of mass 2.8 kg moves at 19.8 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 19.8^2 and multiply by 0.5 * 2.8.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=60 degrees through the sine of 2 theta.
3. Substitute v=59.8 m/s and g=9.81 m/s^2.
Final: {"answer": 315.6923, "unit": "m"}

Answer: 315.6923

Prompt: A projectile is launched on level ground with no air resistance at speed 59.8 m/s and angle 60 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=60 degrees through the sine of 2 theta.
Substitute v=59.8 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=146.1 V and R=82.3 ohm.
Final: {"answer": 259.3586, "unit": "W"}

Answer: 259.3586

Prompt: A resistor has resistance 82.3 ohm and voltage 146.1 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=146.1 V and R=82.3 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=52.5 mg, t=151.5 h, and T_half=105 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 19.3115, "unit": "mg"}

Answer: 19.3115

Prompt: A radioactive sample starts with 52.5 mg. Its half-life is 105 hours. How many mg remain after 151.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=52.5 mg, t=151.5 h, and T_half=105 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 4.57 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 82.3286, "unit": "g"}

Answer: 82.3286

Prompt: A sample contains 4.57 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 4.57 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2135 mL to 2.135 L.
3. Compute M = 4.26 / 2.135.
Final: {"answer": 1.9953, "unit": "mol/L"}

Answer: 1.9953

Prompt: A solution contains 4.26 mol of solute in 2135 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2135 mL to 2.135 L.
Compute M = 4.26 / 2.135.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(2.8)=-2.9 and f(9)=-2.9.
3. The interval width is 6.2.
Final: {"answer": -17.98, "unit": "area_units"}

Answer: -17.98

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=0x+-2.9 from x=2.8 to x=9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(2.8)=-2.9 and f(9)=-2.9.
The interval width is 6.2.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 4309.0 / 100.
Final: {"answer": 43.09, "unit": "m"}

Answer: 43.09

Prompt: Convert 4309 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 4309.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=75 degrees and keep the unit as newtons.
Final: {"answer": [107.7981, 402.3081], "unit": "N"}

Answer: 107.7981

Answer: 402.3081

Prompt: A vector has magnitude 416.5 N at 75 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=75 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.65, 29.0], "unit": "dimensionless"}

Answer: 8.65

Answer: 29.0

Prompt: A line passes through points (1.75, 44.1375) and (15.5, 163.075). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. volt squared divided by ohm is watt
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In instrument manual case 57058, check dimensional consistency for P = V^2 / R. The left side has dimensions W and the right side has dimensions W. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
volt squared divided by ohm is watt
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: recurrence, objective, tie_breaking_rule.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["recurrence", "objective", "tie_breaking_rule"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a routing scoring problem with 1521 items and budget 17985, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: recurrence, objective, tie_breaking_rule.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 18 kg and a = 11.85 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 213.3, "unit": "N"}

Answer: 213.3

Prompt: A cart has mass 18 kg and measured acceleration 11.85 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 18 kg and a = 11.85 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 36.5^2 and multiply by 0.5 * 20.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 13489.0312, "unit": "J"}

Answer: 13489.0312

Prompt: A body of mass 20.25 kg moves at 36.5 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 36.5^2 and multiply by 0.5 * 20.25.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=26 degrees through the sine of 2 theta.
3. Substitute v=79.4 m/s and g=9.81 m/s^2.
Final: {"answer": 506.4122, "unit": "m"}

Answer: 506.4122

Prompt: A projectile is launched on level ground with no air resistance at speed 79.4 m/s and angle 26 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=26 degrees through the sine of 2 theta.
Substitute v=79.4 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 2.76 * 8.314 * 387 / 13.4.
Final: {"answer": 662.7127, "unit": "kPa"}

Answer: 662.7127

Prompt: An ideal gas sample has n=2.76 mol, T=387 K, and V=13.4 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2.76 * 8.314 * 387 / 13.4.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1225 * 4.195 * 1.
Final: {"answer": 5138.875, "unit": "J"}

Answer: 5138.875

Prompt: A 1225 g sample has specific heat 4.195 J/(g*K). How much heat is needed to raise its temperature by 1 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1225 * 4.195 * 1.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=13.9 V and R=165.3 ohm.
Final: {"answer": 1.1688, "unit": "W"}

Answer: 1.1688

Prompt: A resistor has resistance 165.3 ohm and voltage 13.9 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=13.9 V and R=165.3 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=556 mg, t=53.5 h, and T_half=104 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 389.2405, "unit": "mg"}

Answer: 389.2405

Prompt: A radioactive sample starts with 556 mg. Its half-life is 104 hours. How many mg remain after 53.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=556 mg, t=53.5 h, and T_half=104 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 1920 mL to 1.92 L.
3. Compute M = 0.495 / 1.92.
Final: {"answer": 0.2578, "unit": "mol/L"}

Answer: 0.2578

Prompt: A solution contains 0.495 mol of solute in 1920 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1920 mL to 1.92 L.
Compute M = 0.495 / 1.92.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.18 and add it to the current value.
Final: {"answer": [1.346, 1.723, 2.065], "unit": "temperature_units"}

Answer: 1.346

Answer: 1.723

Answer: 2.065

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-2, right=0.75; current interior values are [2.3, 1.3, 2.65]. Use r=0.18. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.18 and add it to the current value.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 4295.0 / 100.
Final: {"answer": 42.95, "unit": "m"}

Answer: 42.95

Prompt: Convert 4295 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 4295.0 / 100.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-9.05, 26.25], "unit": "dimensionless"}

Answer: -9.05

Answer: 26.25

Prompt: A line passes through points (-13.75, 150.688) and (-6.75, 87.3375). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. speed squared over acceleration has units of meters
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In sensor calibration case 563286, check dimensional consistency for R = v^2 * sin(theta) / g. The left side has dimensions m and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
speed squared over acceleration has units of meters
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: volume.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["volume"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute pressure from the ideal gas law. The prompt gives n=0.72 mol and T=274 K, but does not give volume. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: volume.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 6.65 kg and a = 14.75 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 98.0875, "unit": "N"}

Answer: 98.0875

Prompt: A cart has mass 6.65 kg and measured acceleration 14.75 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 6.65 kg and a = 14.75 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 1.35^2 and multiply by 0.5 * 7.75.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 7.0622, "unit": "J"}

Answer: 7.0622

Prompt: A body of mass 7.75 kg moves at 1.35 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 1.35^2 and multiply by 0.5 * 7.75.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=47 degrees through the sine of 2 theta.
3. Substitute v=46.8 m/s and g=9.81 m/s^2.
Final: {"answer": 222.7222, "unit": "m"}

Answer: 222.7222

Prompt: A projectile is launched on level ground with no air resistance at speed 46.8 m/s and angle 47 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=47 degrees through the sine of 2 theta.
Substitute v=46.8 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.3 * 8.314 * 568 / 89.7.
Final: {"answer": 226.3781, "unit": "kPa"}

Answer: 226.3781

Prompt: An ideal gas sample has n=4.3 mol, T=568 K, and V=89.7 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.3 * 8.314 * 568 / 89.7.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 475 * 2.425 * 68.
Final: {"answer": 78327.5, "unit": "J"}

Answer: 78327.5

Prompt: A 475 g sample has specific heat 2.425 J/(g*K). How much heat is needed to raise its temperature by 68 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 475 * 2.425 * 68.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=125.5 V and R=137.5 ohm.
Final: {"answer": 114.5473, "unit": "W"}

Answer: 114.5473

Prompt: A resistor has resistance 137.5 ohm and voltage 125.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=125.5 V and R=137.5 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=546.5 mg, t=35.5 h, and T_half=83 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 406.2902, "unit": "mg"}

Answer: 406.2902

Prompt: A radioactive sample starts with 546.5 mg. Its half-life is 83 hours. How many mg remain after 35.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=546.5 mg, t=35.5 h, and T_half=83 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 6.02 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 264.9402, "unit": "g"}

Answer: 264.9402

Prompt: A sample contains 6.02 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.02 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 1605 mL to 1.605 L.
3. Compute M = 2.13 / 1.605.
Final: {"answer": 1.3271, "unit": "mol/L"}

Answer: 1.3271

Prompt: A solution contains 2.13 mol of solute in 1605 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1605 mL to 1.605 L.
Compute M = 2.13 / 1.605.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.35 y, so each step multiplies y by 1 - h*0.35.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": 1.7923, "unit": "state_units"}

Answer: 1.7923

Prompt: Use explicit Euler on dy/dt = -0.35 y with y(0)=56, step size h=1.95, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.35 y, so each step multiplies y by 1 - h*0.35.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-2.8)=9.62 and f(0.2)=3.92.
3. The interval width is 3.
Final: {"answer": 20.31, "unit": "area_units"}

Answer: 20.31

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-1.9x+4.3 from x=-2.8 to x=0.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-2.8)=9.62 and f(0.2)=3.92.
The interval width is 3.


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 167.5 + 273.15.
Final: {"answer": 440.65, "unit": "K"}

Answer: 440.65

Prompt: Convert 167.5 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute 167.5 + 273.15.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=223 degrees and keep the unit as newtons.
Final: {"answer": [-300.952, -280.6423], "unit": "N"}

Answer: -300.952

Answer: -280.6423

Prompt: A vector has magnitude 411.5 N at 223 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=223 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [1.8, 20.25], "unit": "dimensionless"}

Answer: 1.8

Answer: 20.25

Prompt: A line passes through points (-10.75, 0.9) and (4.25, 27.9). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. speed squared over acceleration has units of meters
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In field worksheet case 741078, check dimensional consistency for R = v^2 * sin(theta) / g. The left side has dimensions m and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
speed squared over acceleration has units of meters
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def normalize_005254(values):
    total = sum(values)
    if total == 0:
        raise ValueError('cannot normalize zero-sum values')
    return [v / total for v in values]
```
Final: {"answer": "python_code", "function": "normalize_005254"}

Answer: python_code

Prompt: Write a Python function normalize_005254(values) that returns a new list whose values sum to 1. Raise ValueError if the sum is zero. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

out = normalize_005254([2, 3, 5]); assert len(out) == 3

try:
    normalize_005254([1, -1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: recurrence, objective, tie_breaking_rule.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["recurrence", "objective", "tie_breaking_rule"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a knapsack scoring problem with 4808 items and budget 28680, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: recurrence, objective, tie_breaking_rule.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 4.15^2 and multiply by 0.5 * 17.3.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 148.9746, "unit": "J"}

Answer: 148.9746

Prompt: A body of mass 17.3 kg moves at 4.15 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 4.15^2 and multiply by 0.5 * 17.3.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=20 degrees through the sine of 2 theta.
3. Substitute v=46.1 m/s and g=9.81 m/s^2.
Final: {"answer": 139.2516, "unit": "m"}

Answer: 139.2516

Prompt: A projectile is launched on level ground with no air resistance at speed 46.1 m/s and angle 20 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=20 degrees through the sine of 2 theta.
Substitute v=46.1 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 0.76 * 8.314 * 612 / 41.6.
Final: {"answer": 92.9569, "unit": "kPa"}

Answer: 92.9569

Prompt: An ideal gas sample has n=0.76 mol, T=612 K, and V=41.6 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 0.76 * 8.314 * 612 / 41.6.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1650 * 4.09 * 114.
Final: {"answer": 769329.0, "unit": "J"}

Answer: 769329.0

Prompt: A 1650 g sample has specific heat 4.09 J/(g*K). How much heat is needed to raise its temperature by 114 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1650 * 4.09 * 114.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=20.3 V and R=15.6 ohm.
Final: {"answer": 26.416, "unit": "W"}

Answer: 26.416

Prompt: A resistor has resistance 15.6 ohm and voltage 20.3 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=20.3 V and R=15.6 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=429 mg, t=218 h, and T_half=54 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 26.1329, "unit": "mg"}

Answer: 26.1329

Prompt: A radioactive sample starts with 429 mg. Its half-life is 54 hours. How many mg remain after 218 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=429 mg, t=218 h, and T_half=54 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.76 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 13.6914, "unit": "g"}

Answer: 13.6914

Prompt: A sample contains 0.76 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.76 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2105 mL to 2.105 L.
3. Compute M = 4.41 / 2.105.
Final: {"answer": 2.095, "unit": "mol/L"}

Answer: 2.095

Prompt: A solution contains 4.41 mol of solute in 2105 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2105 mL to 2.105 L.
Compute M = 4.41 / 2.105.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.49 y, so each step multiplies y by 1 - h*0.49.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 0.0064, "unit": "state_units"}

Answer: 0.0064

Prompt: Use explicit Euler on dy/dt = -0.49 y with y(0)=58.5, step size h=1.3, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.49 y, so each step multiplies y by 1 - h*0.49.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 37100.0 / 1000.
Final: {"answer": 37.1, "unit": "kg"}

Answer: 37.1

Prompt: Convert 37100 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 37100.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=328 degrees and keep the unit as newtons.
Final: {"answer": [64.0276, -40.0089], "unit": "N"}

Answer: 64.0276

Answer: -40.0089

Prompt: A vector has magnitude 75.5 N at 328 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=328 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [9.0, 18.25], "unit": "dimensionless"}

Answer: 9.0

Answer: 18.25

Prompt: A line passes through points (19.5, 193.75) and (21.75, 214). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. lever arm times force has torque dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In model card equation case 44036, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
lever arm times force has torque dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def normalize_005272(values):
    total = sum(values)
    if total == 0:
        raise ValueError('cannot normalize zero-sum values')
    return [v / total for v in values]
```
Final: {"answer": "python_code", "function": "normalize_005272"}

Answer: python_code

Prompt: Write a Python function normalize_005272(values) that returns a new list whose values sum to 1. Raise ValueError if the sum is zero. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

out = normalize_005272([2, 3, 5]); assert len(out) == 3

try:
    normalize_005272([1, -1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: target_schema, valid_examples.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["target_schema", "valid_examples"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a parser that converts each registry clinical record from batch 711772 into the correct schema, but the prompt does not include the target schema or examples. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: target_schema, valid_examples.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 29.1^2 and multiply by 0.5 * 6.5.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2752.1325, "unit": "J"}

Answer: 2752.1325

Prompt: A body of mass 6.5 kg moves at 29.1 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 29.1^2 and multiply by 0.5 * 6.5.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=66 degrees through the sine of 2 theta.
3. Substitute v=76.3 m/s and g=9.81 m/s^2.
Final: {"answer": 441.0152, "unit": "m"}

Answer: 441.0152

Prompt: A projectile is launched on level ground with no air resistance at speed 76.3 m/s and angle 66 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=66 degrees through the sine of 2 theta.
Substitute v=76.3 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 115 * 1.69 * 6.
Final: {"answer": 1166.1, "unit": "J"}

Answer: 1166.1

Prompt: A 115 g sample has specific heat 1.69 J/(g*K). How much heat is needed to raise its temperature by 6 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 115 * 1.69 * 6.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=47.4 V and R=43 ohm.
Final: {"answer": 52.2502, "unit": "W"}

Answer: 52.2502

Prompt: A resistor has resistance 43 ohm and voltage 47.4 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=47.4 V and R=43 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=195 mg, t=356.5 h, and T_half=71.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 6.1531, "unit": "mg"}

Answer: 6.1531

Prompt: A radioactive sample starts with 195 mg. Its half-life is 71.5 hours. How many mg remain after 356.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=195 mg, t=356.5 h, and T_half=71.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.87 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1417.8277, "unit": "g"}

Answer: 1417.8277

Prompt: A sample contains 7.87 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 7.87 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.61 y, so each step multiplies y by 1 - h*0.61.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 12.8103, "unit": "state_units"}

Answer: 12.8103

Prompt: Use explicit Euler on dy/dt = -0.61 y with y(0)=14.5, step size h=0.05, for 4 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.61 y, so each step multiplies y by 1 - h*0.61.
Apply that update for 4 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.2)=-15.46 and f(0.9)=-8.83.
3. The interval width is 5.1.
Final: {"answer": -61.9395, "unit": "area_units"}

Answer: -61.9395

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=1.3x+-10 from x=-4.2 to x=0.9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.2)=-15.46 and f(0.9)=-8.83.
The interval width is 5.1.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.28 and add it to the current value.
Final: {"answer": [0.54, 0.512, 0.542], "unit": "temperature_units"}

Answer: 0.54

Answer: 0.512

Answer: 0.542

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.15, right=2; current interior values are [2.15, -0.3, 0.15]. Use r=0.28. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.28 and add it to the current value.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 47206.0 / 1000.
Final: {"answer": 47.206, "unit": "kg"}

Answer: 47.206

Prompt: Convert 47206 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 47206.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=353 degrees and keep the unit as newtons.
Final: {"answer": [391.5595, -48.0775], "unit": "N"}

Answer: 391.5595

Answer: -48.0775

Prompt: A vector has magnitude 394.5 N at 353 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=353 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. m*v is momentum, not energy
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In sensor calibration case 863758, check dimensional consistency for E = m * v. The left side has dimensions kg*m^2/s^2 and the right side has dimensions kg*m/s. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
m*v is momentum, not energy
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 12.5 kg and a = 8.55 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 106.875, "unit": "N"}

Answer: 106.875

Prompt: A cart has mass 12.5 kg and measured acceleration 8.55 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 12.5 kg and a = 8.55 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 23.25^2 and multiply by 0.5 * 14.75.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 3986.6484, "unit": "J"}

Answer: 3986.6484

Prompt: A body of mass 14.75 kg moves at 23.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 23.25^2 and multiply by 0.5 * 14.75.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=33 degrees through the sine of 2 theta.
3. Substitute v=45.1 m/s and g=9.81 m/s^2.
Final: {"answer": 189.4149, "unit": "m"}

Answer: 189.4149

Prompt: A projectile is launched on level ground with no air resistance at speed 45.1 m/s and angle 33 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=33 degrees through the sine of 2 theta.
Substitute v=45.1 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.22 * 8.314 * 438 / 40.2.
Final: {"answer": 472.8556, "unit": "kPa"}

Answer: 472.8556

Prompt: An ideal gas sample has n=5.22 mol, T=438 K, and V=40.2 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 5.22 * 8.314 * 438 / 40.2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 715 * 2.59 * 115.
Final: {"answer": 212962.75, "unit": "J"}

Answer: 212962.75

Prompt: A 715 g sample has specific heat 2.59 J/(g*K). How much heat is needed to raise its temperature by 115 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 715 * 2.59 * 115.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=73.1 V and R=29.4 ohm.
Final: {"answer": 181.7554, "unit": "W"}

Answer: 181.7554

Prompt: A resistor has resistance 29.4 ohm and voltage 73.1 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=73.1 V and R=29.4 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=51 mg, t=26.5 h, and T_half=42.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 33.1031, "unit": "mg"}

Answer: 33.1031

Prompt: A radioactive sample starts with 51 mg. Its half-life is 42.5 hours. How many mg remain after 26.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=51 mg, t=26.5 h, and T_half=42.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.78 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 78.0679, "unit": "g"}

Answer: 78.0679

Prompt: A sample contains 0.78 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.78 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3995 mL to 3.995 L.
3. Compute M = 3.07 / 3.995.
Final: {"answer": 0.7685, "unit": "mol/L"}

Answer: 0.7685

Prompt: A solution contains 3.07 mol of solute in 3995 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3995 mL to 3.995 L.
Compute M = 3.07 / 3.995.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.61 y, so each step multiplies y by 1 - h*0.61.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": -0.0001, "unit": "state_units"}

Answer: -0.0001

Prompt: Use explicit Euler on dy/dt = -0.61 y with y(0)=20, step size h=1.9, for 7 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.61 y, so each step multiplies y by 1 - h*0.61.
Apply that update for 7 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(0.6)=9.72 and f(7.8)=-6.84.
3. The interval width is 7.2.
Final: {"answer": 10.368, "unit": "area_units"}

Answer: 10.368

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-2.3x+11.1 from x=0.6 to x=7.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(0.6)=9.72 and f(7.8)=-6.84.
The interval width is 7.2.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.3 and add it to the current value.
Final: {"answer": [1.075, 0.355, -0.035], "unit": "temperature_units"}

Answer: 1.075

Answer: 0.355

Answer: -0.035

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.75, right=-0.15; current interior values are [0.4, 1.3, -0.95]. Use r=0.3. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.3 and add it to the current value.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 995.5 / 100.
Final: {"answer": 9.955, "unit": "m"}

Answer: 9.955

Prompt: Convert 995.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 995.5 / 100.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-0.8, 0.5], "unit": "dimensionless"}

Answer: -0.8

Answer: 0.5

Prompt: A line passes through points (-18.75, 15.5) and (-4.25, 3.9). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. amount times volume is not concentration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In model card equation case 701523, check dimensional consistency for C = n * V. The left side has dimensions mol/L and the right side has dimensions mol*L. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
amount times volume is not concentration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def clamp_005308(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_005308"}

Answer: python_code

Prompt: Write a Python function clamp_005308(x, lo, hi) that clips x into the inclusive interval [lo, hi]. It must raise ValueError if lo > hi. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    clamp_005308(1, 3, 2)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=19 degrees through the sine of 2 theta.
3. Substitute v=49.9 m/s and g=9.81 m/s^2.
Final: {"answer": 156.2694, "unit": "m"}

Answer: 156.2694

Prompt: A projectile is launched on level ground with no air resistance at speed 49.9 m/s and angle 19 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=19 degrees through the sine of 2 theta.
Substitute v=49.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.63 * 8.314 * 511 / 116.5.
Final: {"answer": 132.3767, "unit": "kPa"}

Answer: 132.3767

Prompt: An ideal gas sample has n=3.63 mol, T=511 K, and V=116.5 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 3.63 * 8.314 * 511 / 116.5.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=11.2 V and R=160 ohm.
Final: {"answer": 0.784, "unit": "W"}

Answer: 0.784

Prompt: A resistor has resistance 160 ohm and voltage 11.2 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=11.2 V and R=160 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=717 mg, t=353.5 h, and T_half=33.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.4775, "unit": "mg"}

Answer: 0.4775

Prompt: A radioactive sample starts with 717 mg. Its half-life is 33.5 hours. How many mg remain after 353.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=717 mg, t=353.5 h, and T_half=33.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 5.19 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 303.3036, "unit": "g"}

Answer: 303.3036

Prompt: A sample contains 5.19 mol of sodium chloride. Using molar mass 58.44 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 5.19 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.61 y, so each step multiplies y by 1 - h*0.61.
3. Apply that update for 10 steps and round the final state.
Final: {"answer": 0.1698, "unit": "state_units"}

Answer: 0.1698

Prompt: Use explicit Euler on dy/dt = -0.61 y with y(0)=44.5, step size h=0.7, for 10 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.61 y, so each step multiplies y by 1 - h*0.61.
Apply that update for 10 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-0.6)=15.94 and f(0.7)=20.62.
3. The interval width is 1.3.
Final: {"answer": 23.764, "unit": "area_units"}

Answer: 23.764

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=3.6x+18.1 from x=-0.6 to x=0.7. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-0.6)=15.94 and f(0.7)=20.62.
The interval width is 1.3.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=220 degrees and keep the unit as newtons.
Final: {"answer": [-52.091, -43.7096], "unit": "N"}

Answer: -52.091

Answer: -43.7096

Prompt: A vector has magnitude 68 N at 220 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=220 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-0.6, 4.25], "unit": "dimensionless"}

Answer: -0.6

Answer: 4.25

Prompt: A line passes through points (-12, 11.45) and (0, 4.25). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. amount times volume is not concentration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In instrument manual case 345739, check dimensional consistency for C = n * V. The left side has dimensions mol/L and the right side has dimensions mol*L. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
amount times volume is not concentration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: recurrence, objective, tie_breaking_rule.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["recurrence", "objective", "tie_breaking_rule"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a scheduler scoring problem with 4349 items and budget 40045, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: recurrence, objective, tie_breaking_rule.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 3 kg and a = 7.9 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 23.7, "unit": "N"}

Answer: 23.7

Prompt: A cart has mass 3 kg and measured acceleration 7.9 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 3 kg and a = 7.9 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=36 degrees through the sine of 2 theta.
3. Substitute v=60.5 m/s and g=9.81 m/s^2.
Final: {"answer": 354.8527, "unit": "m"}

Answer: 354.8527

Prompt: A projectile is launched on level ground with no air resistance at speed 60.5 m/s and angle 36 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=36 degrees through the sine of 2 theta.
Substitute v=60.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 2.22 * 8.314 * 561 / 85.7.
Final: {"answer": 120.8217, "unit": "kPa"}

Answer: 120.8217

Prompt: An ideal gas sample has n=2.22 mol, T=561 K, and V=85.7 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2.22 * 8.314 * 561 / 85.7.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1490 * 3.76 * 95.
Final: {"answer": 532228.0, "unit": "J"}

Answer: 532228.0

Prompt: A 1490 g sample has specific heat 3.76 J/(g*K). How much heat is needed to raise its temperature by 95 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1490 * 3.76 * 95.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=765.5 mg, t=212.5 h, and T_half=59.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 64.3929, "unit": "mg"}

Answer: 64.3929

Prompt: A radioactive sample starts with 765.5 mg. Its half-life is 59.5 hours. How many mg remain after 212.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=765.5 mg, t=212.5 h, and T_half=59.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 4.79 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 210.8079, "unit": "g"}

Answer: 210.8079

Prompt: A sample contains 4.79 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 4.79 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2135 mL to 2.135 L.
3. Compute M = 1.655 / 2.135.
Final: {"answer": 0.7752, "unit": "mol/L"}

Answer: 0.7752

Prompt: A solution contains 1.655 mol of solute in 2135 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2135 mL to 2.135 L.
Compute M = 1.655 / 2.135.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.33 y, so each step multiplies y by 1 - h*0.33.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 1.3728, "unit": "state_units"}

Answer: 1.3728

Prompt: Use explicit Euler on dy/dt = -0.33 y with y(0)=79.5, step size h=1.1, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.33 y, so each step multiplies y by 1 - h*0.33.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-1.2)=-3.5 and f(3.5)=24.7.
3. The interval width is 4.7.
Final: {"answer": 49.82, "unit": "area_units"}

Answer: 49.82

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=6x+3.7 from x=-1.2 to x=3.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-1.2)=-3.5 and f(3.5)=24.7.
The interval width is 4.7.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.4 and add it to the current value.
Final: {"answer": [0.13, 1.63, 0.87], "unit": "temperature_units"}

Answer: 0.13

Answer: 1.63

Answer: 0.87

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.6, right=-0.3; current interior values are [1.15, 1.35, 2.25]. Use r=0.4. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.4 and add it to the current value.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 1468.5 / 100.
Final: {"answer": 14.685, "unit": "m"}

Answer: 14.685

Prompt: Convert 1468.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 1468.5 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=10 degrees and keep the unit as newtons.
Final: {"answer": [219.6121, 38.7235], "unit": "N"}

Answer: 219.6121

Answer: 38.7235

Prompt: A vector has magnitude 223 N at 10 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=10 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [2.3, -22.75], "unit": "dimensionless"}

Answer: 2.3

Answer: -22.75

Prompt: A line passes through points (20, 23.25) and (27.25, 39.925). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.