Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 11.65 kg and a = 7.75 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 90.2875, "unit": "N"}

Answer: 90.2875

Prompt: A cart has mass 11.65 kg and measured acceleration 7.75 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 = 11.65 kg and a = 7.75 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.56 y, so each step multiplies y by 1 - h*0.56.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 19.0804, "unit": "state_units"}

Answer: 19.0804

Prompt: Use explicit Euler on dy/dt = -0.56 y with y(0)=71, step size h=0.5, 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) = -0.56 y, so each step multiplies y by 1 - h*0.56.
Apply that update for 4 steps and round the final state.


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": [-4.7, 19.25], "unit": "dimensionless"}

Answer: -4.7

Answer: 19.25

Prompt: A line passes through points (-10, 66.25) and (2.25, 8.675). 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. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-2.75, -5.0], "unit": "dimensionless"}

Answer: -2.75

Answer: -5.0

Prompt: A line passes through points (15.5, -47.625) and (23, -68.25). 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. 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 sensor calibration case 253066, 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 x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=144 degrees and keep the unit as newtons.
Final: {"answer": [-328.4609, 238.6408], "unit": "N"}

Answer: -328.4609

Answer: 238.6408

Prompt: A vector has magnitude 406 N at 144 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=144 degrees and keep the unit as newtons.

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


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 = 3.62 * 8.314 * 299 / 1.9.
Final: {"answer": 4736.267, "unit": "kPa"}

Answer: 4736.267

Prompt: An ideal gas sample has n=3.62 mol, T=299 K, and V=1.9 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 = 3.62 * 8.314 * 299 / 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=0.3 and add it to the current value.
Final: {"answer": [1.685, 1.36, 1.11], "unit": "temperature_units"}

Answer: 1.685

Answer: 1.36

Answer: 1.11

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=1.9; current interior values are [2.6, 1.0, 0.6]. 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. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 165.5 + 273.15.
Final: {"answer": 438.65, "unit": "K"}

Answer: 438.65

Prompt: Convert 165.5 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 165.5 + 273.15.


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 = 5.55 * 8.314 * 392 / 44.1.
Final: {"answer": 410.1573, "unit": "kPa"}

Answer: 410.1573

Prompt: An ideal gas sample has n=5.55 mol, T=392 K, and V=44.1 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 = 5.55 * 8.314 * 392 / 44.1.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(9.1)=-25.27 and f(18.4)=-59.68.
3. The interval width is 9.3.
Final: {"answer": -395.0175, "unit": "area_units"}

Answer: -395.0175

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.7x+8.4 from x=9.1 to x=18.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(9.1)=-25.27 and f(18.4)=-59.68.
The interval width is 9.3.

(x1 - x0) * ((slope * x0 + intercept) + (slope * x1 + intercept)) / 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 = 4.71 * 8.314 * 488 / 67.7.
Final: {"answer": 282.2683, "unit": "kPa"}

Answer: 282.2683

Prompt: An ideal gas sample has n=4.71 mol, T=488 K, and V=67.7 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 = 4.71 * 8.314 * 488 / 67.7.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-5.6)=42.3 and f(-1.7)=24.75.
3. The interval width is 3.9.
Final: {"answer": 130.7475, "unit": "area_units"}

Answer: 130.7475

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-4.5x+17.1 from x=-5.6 to x=-1.7. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-5.6)=42.3 and f(-1.7)=24.75.
The interval width is 3.9.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1900 * 3.44 * 82.
Final: {"answer": 535952.0, "unit": "J"}

Answer: 535952.0

Prompt: A 1900 g sample has specific heat 3.44 J/(g*K). How much heat is needed to raise its temperature by 82 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1900 * 3.44 * 82.


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_004066(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_004066"}

Answer: python_code

Prompt: Write a Python function normalize_004066(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_004066([2, 3, 5]); assert len(out) == 3

try:
    normalize_004066([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 scheduler scoring problem with 3968 items and budget 79375, 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. Mass equals amount in moles times molar mass.
2. Use m = 1.03 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 45.3303, "unit": "g"}

Answer: 45.3303

Prompt: A sample contains 1.03 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 = 1.03 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


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=0.38 and add it to the current value.
Final: {"answer": [1.263, 0.961, 1.837], "unit": "temperature_units"}

Answer: 1.263

Answer: 0.961

Answer: 1.837

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=2.85, right=2.75; current interior values are [-0.2, 0.6, 2.35]. Use r=0.38. 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=0.38 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=358 degrees and keep the unit as newtons.
Final: {"answer": [418.7448, -14.6229], "unit": "N"}

Answer: 418.7448

Answer: -14.6229

Prompt: A vector has magnitude 419 N at 358 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=358 degrees and keep the unit as newtons.


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

Answer: INSUFFICIENT_INFO

Prompt: Solve the 1D heat equation after one finite-difference step. The prompt gives r=0.14 and interior values [1.15, -1.85, 1.5], but gives no boundary values. 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: left_boundary, right_boundary.
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 = 23^2 and multiply by 0.5 * 1.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 330.625, "unit": "J"}

Answer: 330.625

Prompt: A body of mass 1.25 kg moves at 23 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 23^2 and multiply by 0.5 * 1.25.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2060 mL to 2.06 L.
3. Compute M = 4.235 / 2.06.
Final: {"answer": 2.0558, "unit": "mol/L"}

Answer: 2.0558

Prompt: A solution contains 4.235 mol of solute in 2060 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2060 mL to 2.06 L.
Compute M = 4.235 / 2.06.


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": [1.089, -0.657, 1.47], "unit": "temperature_units"}

Answer: 1.089

Answer: -0.657

Answer: 1.47

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=-0.55, right=1.4; current interior values are [1.95, -1.7, 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. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 32.7^2 and multiply by 0.5 * 10.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 5480.1113, "unit": "J"}

Answer: 5480.1113

Prompt: A body of mass 10.25 kg moves at 32.7 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 32.7^2 and multiply by 0.5 * 10.25.
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=33 degrees through the sine of 2 theta.
3. Substitute v=13.7 m/s and g=9.81 m/s^2.
Final: {"answer": 17.4784, "unit": "m"}

Answer: 17.4784

Prompt: A projectile is launched on level ground with no air resistance at speed 13.7 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=13.7 m/s and g=9.81 m/s^2.

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


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 1211.0 / 100.
Final: {"answer": 12.11, "unit": "m"}

Answer: 12.11

Prompt: Convert 1211 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 1211.0 / 100.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=460.5 mg, t=312 h, and T_half=44 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 3.378, "unit": "mg"}

Answer: 3.378

Prompt: A radioactive sample starts with 460.5 mg. Its half-life is 44 hours. How many mg remain after 312 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=460.5 mg, t=312 h, and T_half=44 h.
The remaining amount has the same mass unit as the initial amount.

initial_amount * 0.5 ** (elapsed_time / half_life)


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.02 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 308.9502, "unit": "g"}

Answer: 308.9502

Prompt: A sample contains 7.02 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 = 7.02 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


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=118.1 V and R=42.2 ohm.
Final: {"answer": 330.5121, "unit": "W"}

Answer: 330.5121

Prompt: A resistor has resistance 42.2 ohm and voltage 118.1 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=118.1 V and R=42.2 ohm.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.98 y, so each step multiplies y by 1 - h*0.98.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 0.0, "unit": "state_units"}

Answer: 0.0

Prompt: Use explicit Euler on dy/dt = -0.98 y with y(0)=33.5, step size h=0.9, 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) = -0.98 y, so each step multiplies y by 1 - h*0.98.
Apply that update for 9 steps and round the final state.


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_004174(values):
    total = 0
    out = []
    for value in values:
        total += value
        out.append(total)
    return out
```
Final: {"answer": "python_code", "function": "running_total_004174"}

Answer: python_code

Prompt: Write a Python function running_total_004174(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. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.59 y, so each step multiplies y by 1 - h*0.59.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 2.9247, "unit": "state_units"}

Answer: 2.9247

Prompt: Use explicit Euler on dy/dt = -0.59 y with y(0)=56.5, step size h=0.4, for 11 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.59 y, so each step multiplies y by 1 - h*0.59.
Apply that update for 11 steps and round the final state.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=937 mg, t=147 h, and T_half=87.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 292.4214, "unit": "mg"}

Answer: 292.4214

Prompt: A radioactive sample starts with 937 mg. Its half-life is 87.5 hours. How many mg remain after 147 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=937 mg, t=147 h, and T_half=87.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(9.3)=-17.27 and f(18.4)=-52.76.
3. The interval width is 9.1.
Final: {"answer": -318.6365, "unit": "area_units"}

Answer: -318.6365

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.9x+19 from x=9.3 to x=18.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(9.3)=-17.27 and f(18.4)=-52.76.
The interval width is 9.1.


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 251768, 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. Use Newton's second law F = m a.
2. Substitute m = 7.25 kg and a = 1.1 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 7.975, "unit": "N"}

Answer: 7.975

Prompt: A cart has mass 7.25 kg and measured acceleration 1.1 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 = 7.25 kg and a = 1.1 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 = 2.99 * 8.314 * 618 / 11.8.
Final: {"answer": 1301.9301, "unit": "kPa"}

Answer: 1301.9301

Prompt: An ideal gas sample has n=2.99 mol, T=618 K, and V=11.8 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 = 2.99 * 8.314 * 618 / 11.8.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 95 * 3.995 * 24.
Final: {"answer": 9108.6, "unit": "J"}

Answer: 9108.6

Prompt: A 95 g sample has specific heat 3.995 J/(g*K). How much heat is needed to raise its temperature by 24 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 95 * 3.995 * 24.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.86 y, so each step multiplies y by 1 - h*0.86.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 0.1308, "unit": "state_units"}

Answer: 0.1308

Prompt: Use explicit Euler on dy/dt = -0.86 y with y(0)=75.5, step size h=1.4, 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) = -0.86 y, so each step multiplies y by 1 - h*0.86.
Apply that update for 4 steps and round the final state.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 27.1^2 and multiply by 0.5 * 19.5.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 7160.4975, "unit": "J"}

Answer: 7160.4975

Prompt: A body of mass 19.5 kg moves at 27.1 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 27.1^2 and multiply by 0.5 * 19.5.
The unit kg*m^2/s^2 is one joule.


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=7.4 V and R=115.2 ohm.
Final: {"answer": 0.4753, "unit": "W"}

Answer: 0.4753

Prompt: A resistor has resistance 115.2 ohm and voltage 7.4 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=7.4 V and R=115.2 ohm.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 4881.0 / 100.
Final: {"answer": 48.81, "unit": "m"}

Answer: 48.81

Prompt: Convert 4881 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 4881.0 / 100.


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": [1.838, 1.315, -0.276], "unit": "temperature_units"}

Answer: 1.838

Answer: 1.315

Answer: -0.276

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=0.8, right=-1.3; current interior values are [2.3, 1.7, -0.65]. 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. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [4.55, 14.5], "unit": "dimensionless"}

Answer: 4.55

Answer: 14.5

Prompt: A line passes through points (-14.5, -51.475) and (-0.25, 13.3625). 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. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=52 degrees through the sine of 2 theta.
3. Substitute v=75.1 m/s and g=9.81 m/s^2.
Final: {"answer": 557.8469, "unit": "m"}

Answer: 557.8469

Prompt: A projectile is launched on level ground with no air resistance at speed 75.1 m/s and angle 52 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=52 degrees through the sine of 2 theta.
Substitute v=75.1 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 230.0 + 273.15.
Final: {"answer": 503.15, "unit": "K"}

Answer: 503.15

Prompt: Convert 230 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 230.0 + 273.15.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.93 y, so each step multiplies y by 1 - h*0.93.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 24.1039, "unit": "state_units"}

Answer: 24.1039

Prompt: Use explicit Euler on dy/dt = -0.93 y with y(0)=37, step size h=0.05, 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) = -0.93 y, so each step multiplies y by 1 - h*0.93.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 1.9 kg and a = 0.5 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 0.95, "unit": "N"}

Answer: 0.95

Prompt: A cart has mass 1.9 kg and measured acceleration 0.5 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 = 1.9 kg and a = 0.5 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1690 * 3.33 * 52.
Final: {"answer": 292640.4, "unit": "J"}

Answer: 292640.4

Prompt: A 1690 g sample has specific heat 3.33 J/(g*K). How much heat is needed to raise its temperature by 52 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1690 * 3.33 * 52.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.56 y, so each step multiplies y by 1 - h*0.56.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 0.0003, "unit": "state_units"}

Answer: 0.0003

Prompt: Use explicit Euler on dy/dt = -0.56 y with y(0)=71.5, step size h=1.2, for 11 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.56 y, so each step multiplies y by 1 - h*0.56.
Apply that update for 11 steps and round the final state.


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 pairwise_ratios_004408(numerators, denominators):
    if len(numerators) != len(denominators):
        raise ValueError('length mismatch')
    out = []
    for numerator, denominator in zip(numerators, denominators):
        if denominator == 0:
            raise ValueError('zero denominator')
        out.append(numerator / denominator)
    return out
```
Final: {"answer": "python_code", "function": "pairwise_ratios_004408"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_004408(numerators, denominators) that returns a list of n/d values. Raise ValueError if lengths differ or any denominator 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.

try:
    pairwise_ratios_004408([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_004408([1, 2], [1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=99 degrees and keep the unit as newtons.
Final: {"answer": [-2.1901, 13.8276], "unit": "N"}

Answer: -2.1901

Answer: 13.8276

Prompt: A vector has magnitude 14 N at 99 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=99 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 2.1 kg and a = 0.6 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 1.26, "unit": "N"}

Answer: 1.26

Prompt: A cart has mass 2.1 kg and measured acceleration 0.6 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 = 2.1 kg and a = 0.6 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 = 5.39 * 8.314 * 280 / 104.9.
Final: {"answer": 119.6138, "unit": "kPa"}

Answer: 119.6138

Prompt: An ideal gas sample has n=5.39 mol, T=280 K, and V=104.9 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 = 5.39 * 8.314 * 280 / 104.9.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 17.95^2 and multiply by 0.5 * 7.15.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1151.8739, "unit": "J"}

Answer: 1151.8739

Prompt: A body of mass 7.15 kg moves at 17.95 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 17.95^2 and multiply by 0.5 * 7.15.
The unit kg*m^2/s^2 is one joule.


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=66.3 V and R=124.5 ohm.
Final: {"answer": 35.3067, "unit": "W"}

Answer: 35.3067

Prompt: A resistor has resistance 124.5 ohm and voltage 66.3 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=66.3 V and R=124.5 ohm.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=140 degrees and keep the unit as newtons.
Final: {"answer": [-252.7947, 212.1199], "unit": "N"}

Answer: -252.7947

Answer: 212.1199

Prompt: A vector has magnitude 330 N at 140 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=140 degrees and keep the unit as newtons.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=55 degrees through the sine of 2 theta.
3. Substitute v=78.9 m/s and g=9.81 m/s^2.
Final: {"answer": 596.3082, "unit": "m"}

Answer: 596.3082

Prompt: A projectile is launched on level ground with no air resistance at speed 78.9 m/s and angle 55 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=55 degrees through the sine of 2 theta.
Substitute v=78.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=209 degrees and keep the unit as newtons.
Final: {"answer": [-143.8749, -79.7512], "unit": "N"}

Answer: -143.8749

Answer: -79.7512

Prompt: A vector has magnitude 164.5 N at 209 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=209 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 17.3^2 and multiply by 0.5 * 18.5.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2768.4325, "unit": "J"}

Answer: 2768.4325

Prompt: A body of mass 18.5 kg moves at 17.3 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 17.3^2 and multiply by 0.5 * 18.5.
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=58 degrees through the sine of 2 theta.
3. Substitute v=76.5 m/s and g=9.81 m/s^2.
Final: {"answer": 536.1842, "unit": "m"}

Answer: 536.1842

Prompt: A projectile is launched on level ground with no air resistance at speed 76.5 m/s and angle 58 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=58 degrees through the sine of 2 theta.
Substitute v=76.5 m/s and g=9.81 m/s^2.


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": [-5.65, 26.0], "unit": "dimensionless"}

Answer: -5.65

Answer: 26.0

Prompt: A line passes through points (10.25, -31.9125) and (17.25, -71.4625). 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. 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 918100, 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 Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 895 * 2.605 * 81.
Final: {"answer": 188849.475, "unit": "J"}

Answer: 188849.475

Prompt: A 895 g sample has specific heat 2.605 J/(g*K). How much heat is needed to raise its temperature by 81 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 895 * 2.605 * 81.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 1190 mL to 1.19 L.
3. Compute M = 0.585 / 1.19.
Final: {"answer": 0.4916, "unit": "mol/L"}

Answer: 0.4916

Prompt: A solution contains 0.585 mol of solute in 1190 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1190 mL to 1.19 L.
Compute M = 0.585 / 1.19.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-6.3)=-28.33 and f(-3.2)=-12.52.
3. The interval width is 3.1.
Final: {"answer": -63.3175, "unit": "area_units"}

Answer: -63.3175

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=5.1x+3.8 from x=-6.3 to x=-3.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-6.3)=-28.33 and f(-3.2)=-12.52.
The interval width is 3.1.


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 quality-control check case 83029, 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
import math

def unit_vector_004588(x, y):
    norm = math.hypot(x, y)
    if norm == 0:
        raise ValueError('zero vector has no direction')
    return (x / norm, y / norm)
```
Final: {"answer": "python_code", "function": "unit_vector_004588"}

Answer: python_code

Prompt: Write a Python function unit_vector_004588(x, y) that returns the 2D unit vector as a tuple. Raise ValueError for the zero vector. 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:
    unit_vector_004588(0, 0)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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=149 V and R=98.6 ohm.
Final: {"answer": 225.1623, "unit": "W"}

Answer: 225.1623

Prompt: A resistor has resistance 98.6 ohm and voltage 149 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=149 V and R=98.6 ohm.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 17.25 kg and a = 4.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 74.175, "unit": "N"}

Answer: 74.175

Prompt: A cart has mass 17.25 kg and measured acceleration 4.3 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 = 17.25 kg and a = 4.3 m/s^2.
The unit kg*m/s^2 is one newton.


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": [2.04, 1.23, 1.14], "unit": "temperature_units"}

Answer: 2.04

Answer: 1.23

Answer: 1.14

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=0.7; current interior values are [2.6, 2.55, -0.8]. 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. Mass equals amount in moles times molar mass.
2. Use m = 0.45 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 19.8045, "unit": "g"}

Answer: 19.8045

Prompt: A sample contains 0.45 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 = 0.45 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-0.4)=12.86 and f(6.6)=-14.44.
3. The interval width is 7.
Final: {"answer": -5.53, "unit": "area_units"}

Answer: -5.53

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.9x+11.3 from x=-0.4 to x=6.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-0.4)=12.86 and f(6.6)=-14.44.
The interval width is 7.


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": [-8.35, -3.0], "unit": "dimensionless"}

Answer: -8.35

Answer: -3.0

Prompt: A line passes through points (-10, 80.5) and (-7.5, 59.625). 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. Molarity is moles of solute divided by solution volume in liters.
2. Convert 40 mL to 0.04 L.
3. Compute M = 0.115 / 0.04.
Final: {"answer": 2.875, "unit": "mol/L"}

Answer: 2.875

Prompt: A solution contains 0.115 mol of solute in 40 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 40 mL to 0.04 L.
Compute M = 0.115 / 0.04.


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

Answer: INSUFFICIENT_INFO

Prompt: Compute projectile range on level ground. The prompt gives initial speed 19.4 m/s but does not give launch angle or vertical velocity. 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: launch_angle_or_vertical_velocity.
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 = 4.45^2 and multiply by 0.5 * 29.85.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 295.5523, "unit": "J"}

Answer: 295.5523

Prompt: A body of mass 29.85 kg moves at 4.45 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 4.45^2 and multiply by 0.5 * 29.85.
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 = 475 * 3.295 * 80.
Final: {"answer": 125210.0, "unit": "J"}

Answer: 125210.0

Prompt: A 475 g sample has specific heat 3.295 J/(g*K). How much heat is needed to raise its temperature by 80 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 475 * 3.295 * 80.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 275 mL to 0.275 L.
3. Compute M = 2.97 / 0.275.
Final: {"answer": 10.8, "unit": "mol/L"}

Answer: 10.8

Prompt: A solution contains 2.97 mol of solute in 275 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 275 mL to 0.275 L.
Compute M = 2.97 / 0.275.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 2420.0 / 100.
Final: {"answer": 24.2, "unit": "m"}

Answer: 24.2

Prompt: Convert 2420 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 2420.0 / 100.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=63 degrees through the sine of 2 theta.
3. Substitute v=68.2 m/s and g=9.81 m/s^2.
Final: {"answer": 383.5813, "unit": "m"}

Answer: 383.5813

Prompt: A projectile is launched on level ground with no air resistance at speed 68.2 m/s and angle 63 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=63 degrees through the sine of 2 theta.
Substitute v=68.2 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=414.5 mg, t=214.5 h, and T_half=39.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 9.6121, "unit": "mg"}

Answer: 9.6121

Prompt: A radioactive sample starts with 414.5 mg. Its half-life is 39.5 hours. How many mg remain after 214.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=414.5 mg, t=214.5 h, and T_half=39.5 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 1935 mL to 1.935 L.
3. Compute M = 2.455 / 1.935.
Final: {"answer": 1.2687, "unit": "mol/L"}

Answer: 1.2687

Prompt: A solution contains 2.455 mol of solute in 1935 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1935 mL to 1.935 L.
Compute M = 2.455 / 1.935.


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=0.1 and add it to the current value.
Final: {"answer": [-1.405, -1.73, -1.355], "unit": "temperature_units"}

Answer: -1.405

Answer: -1.73

Answer: -1.355

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=1.4; current interior values are [-1.65, -1.75, -1.65]. Use r=0.1. 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=0.1 and add it to the current value.


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=0.08 and add it to the current value.
Final: {"answer": [2.072, -0.486, 0.478], "unit": "temperature_units"}

Answer: 2.072

Answer: -0.486

Answer: 0.478

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=2.6, right=1.05; current interior values are [2.3, -0.85, 0.55]. Use r=0.08. 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=0.08 and add it to the current value.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 5.89 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1061.1188, "unit": "g"}

Answer: 1061.1188

Prompt: A sample contains 5.89 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 = 5.89 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=357 degrees and keep the unit as newtons.
Final: {"answer": [393.46, -20.6204], "unit": "N"}

Answer: 393.46

Answer: -20.6204

Prompt: A vector has magnitude 394 N at 357 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=357 degrees and keep the unit as newtons.


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

Answer: consistent

Prompt: In quality-control check case 802997, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
mass times acceleration has newton 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 finite_differences_004804(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_004804"}

Answer: python_code

Prompt: Write a Python function finite_differences_004804(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_004804([5]) == []

assert finite_differences_004804([]) == []


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-3.3)=-25.68 and f(-0.4)=-15.24.
3. The interval width is 2.9.
Final: {"answer": -59.334, "unit": "area_units"}

Answer: -59.334

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=3.6x+-13.8 from x=-3.3 to x=-0.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-3.3)=-25.68 and f(-0.4)=-15.24.
The interval width is 2.9.


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 safe_mean_004822(values):
    if not values:
        raise ValueError('safe_mean requires at least one value')
    return sum(values) / len(values)
```
Final: {"answer": "python_code", "function": "safe_mean_004822"}

Answer: python_code

Prompt: Write a Python function safe_mean_004822(values) that returns the arithmetic mean. It must raise ValueError for an empty input and 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.

try:
    safe_mean_004822([])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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 = 5.4 * 8.314 * 594 / 30.2.
Final: {"answer": 883.0459, "unit": "kPa"}

Answer: 883.0459

Prompt: An ideal gas sample has n=5.4 mol, T=594 K, and V=30.2 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 = 5.4 * 8.314 * 594 / 30.2.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=83 mg, t=191 h, and T_half=98 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 21.4969, "unit": "mg"}

Answer: 21.4969

Prompt: A radioactive sample starts with 83 mg. Its half-life is 98 hours. How many mg remain after 191 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=83 mg, t=191 h, and T_half=98 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(8.5)=51.3 and f(16.1)=86.26.
3. The interval width is 7.6.
Final: {"answer": 522.728, "unit": "area_units"}

Answer: 522.728

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=4.6x+12.2 from x=8.5 to x=16.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(8.5)=51.3 and f(16.1)=86.26.
The interval width is 7.6.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 16.3^2 and multiply by 0.5 * 8.95.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1188.9627, "unit": "J"}

Answer: 1188.9627

Prompt: A body of mass 8.95 kg moves at 16.3 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 16.3^2 and multiply by 0.5 * 8.95.
The unit kg*m^2/s^2 is one joule.


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 = 0.8 * 8.314 * 296 / 94.2.
Final: {"answer": 20.8997, "unit": "kPa"}

Answer: 20.8997

Prompt: An ideal gas sample has n=0.8 mol, T=296 K, and V=94.2 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 = 0.8 * 8.314 * 296 / 94.2.


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_004894(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_004894"}

Answer: python_code

Prompt: Write a Python function finite_differences_004894(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_004894([5]) == []

assert finite_differences_004894([]) == []


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4120 mL to 4.12 L.
3. Compute M = 4.02 / 4.12.
Final: {"answer": 0.9757, "unit": "mol/L"}

Answer: 0.9757

Prompt: A solution contains 4.02 mol of solute in 4120 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4120 mL to 4.12 L.
Compute M = 4.02 / 4.12.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=64 degrees through the sine of 2 theta.
3. Substitute v=84.2 m/s and g=9.81 m/s^2.
Final: {"answer": 569.4916, "unit": "m"}

Answer: 569.4916

Prompt: A projectile is launched on level ground with no air resistance at speed 84.2 m/s and angle 64 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=64 degrees through the sine of 2 theta.
Substitute v=84.2 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.64 y, so each step multiplies y by 1 - h*0.64.
3. Apply that update for 10 steps and round the final state.
Final: {"answer": 0.0, "unit": "state_units"}

Answer: 0.0

Prompt: Use explicit Euler on dy/dt = -0.64 y with y(0)=15.5, step size h=1.75, 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) = -0.64 y, so each step multiplies y by 1 - h*0.64.
Apply that update for 10 steps and round the final state.


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_004930(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_004930"}

Answer: python_code

Prompt: Write a Python function finite_differences_004930(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_004930([5]) == []

assert finite_differences_004930([]) == []


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1160 * 1.39 * 79.
Final: {"answer": 127379.6, "unit": "J"}

Answer: 127379.6

Prompt: A 1160 g sample has specific heat 1.39 J/(g*K). How much heat is needed to raise its temperature by 79 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1160 * 1.39 * 79.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=733.5 mg, t=68 h, and T_half=5.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.1392, "unit": "mg"}

Answer: 0.1392

Prompt: A radioactive sample starts with 733.5 mg. Its half-life is 5.5 hours. How many mg remain after 68 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=733.5 mg, t=68 h, and T_half=5.5 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 4165 mL to 4.165 L.
3. Compute M = 1.07 / 4.165.
Final: {"answer": 0.2569, "unit": "mol/L"}

Answer: 0.2569

Prompt: A solution contains 1.07 mol of solute in 4165 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4165 mL to 4.165 L.
Compute M = 1.07 / 4.165.


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 laboratory experiment record from batch 38771 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. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.89 y, so each step multiplies y by 1 - h*0.89.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": 5.3435, "unit": "state_units"}

Answer: 5.3435

Prompt: Use explicit Euler on dy/dt = -0.89 y with y(0)=47, step size h=0.3, 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) = -0.89 y, so each step multiplies y by 1 - h*0.89.
Apply that update for 7 steps and round the final state.


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_004984(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_004984"}

Answer: python_code

Prompt: Write a Python function center_values_004984(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_004984([])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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=0.42 and add it to the current value.
Final: {"answer": [2.261, 0.698, -0.017], "unit": "temperature_units"}

Answer: 2.261

Answer: 0.698

Answer: -0.017

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=2.85, right=-1.85; current interior values are [1.4, 2.0, -0.5]. Use r=0.42. 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=0.42 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=10 degrees and keep the unit as newtons.
Final: {"answer": [278.2082, 49.0556], "unit": "N"}

Answer: 278.2082

Answer: 49.0556

Prompt: A vector has magnitude 282.5 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": [8.8, 16.5], "unit": "dimensionless"}

Answer: 8.8

Answer: 16.5

Prompt: A line passes through points (-3, -9.9) and (0.5, 20.9). 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. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=180 mg, t=159.5 h, and T_half=16 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.1796, "unit": "mg"}

Answer: 0.1796

Prompt: A radioactive sample starts with 180 mg. Its half-life is 16 hours. How many mg remain after 159.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=180 mg, t=159.5 h, and T_half=16 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=315 degrees and keep the unit as newtons.
Final: {"answer": [9.5459, -9.5459], "unit": "N"}

Answer: 9.5459

Answer: -9.5459

Prompt: A vector has magnitude 13.5 N at 315 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=315 degrees and keep the unit as newtons.


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 safe_mean_005074(values):
    if not values:
        raise ValueError('safe_mean requires at least one value')
    return sum(values) / len(values)
```
Final: {"answer": "python_code", "function": "safe_mean_005074"}

Answer: python_code

Prompt: Write a Python function safe_mean_005074(values) that returns the arithmetic mean. It must raise ValueError for an empty input and 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.

try:
    safe_mean_005074([])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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": [2.08, -0.165, -0.793], "unit": "temperature_units"}

Answer: 2.08

Answer: -0.165

Answer: -0.793

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=-1.4, right=1.35; current interior values are [2.2, -0.2, -0.85]. 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. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=14 degrees through the sine of 2 theta.
3. Substitute v=29.6 m/s and g=9.81 m/s^2.
Final: {"answer": 41.9299, "unit": "m"}

Answer: 41.9299

Prompt: A projectile is launched on level ground with no air resistance at speed 29.6 m/s and angle 14 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=14 degrees through the sine of 2 theta.
Substitute v=29.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.16 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 126.2304, "unit": "g"}

Answer: 126.2304

Prompt: A sample contains 2.16 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 = 2.16 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 3504.5 / 100.
Final: {"answer": 35.045, "unit": "m"}

Answer: 35.045

Prompt: Convert 3504.5 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 3504.5 / 100.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=22 degrees through the sine of 2 theta.
3. Substitute v=81.4 m/s and g=9.81 m/s^2.
Final: {"answer": 469.1925, "unit": "m"}

Answer: 469.1925

Prompt: A projectile is launched on level ground with no air resistance at speed 81.4 m/s and angle 22 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=22 degrees through the sine of 2 theta.
Substitute v=81.4 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-9.6)=18.06 and f(-3.2)=14.22.
3. The interval width is 6.4.
Final: {"answer": 103.296, "unit": "area_units"}

Answer: 103.296

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-0.6x+12.3 from x=-9.6 to x=-3.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-9.6)=18.06 and f(-3.2)=14.22.
The interval width is 6.4.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 8.2^2 and multiply by 0.5 * 18.05.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 606.841, "unit": "J"}

Answer: 606.841

Prompt: A body of mass 18.05 kg moves at 8.2 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 8.2^2 and multiply by 0.5 * 18.05.
The unit kg*m^2/s^2 is one joule.


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 = 4.06 * 8.314 * 394 / 42.3.
Final: {"answer": 314.4068, "unit": "kPa"}

Answer: 314.4068

Prompt: An ideal gas sample has n=4.06 mol, T=394 K, and V=42.3 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 = 4.06 * 8.314 * 394 / 42.3.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 955 * 4.8 * 42.
Final: {"answer": 192528.0, "unit": "J"}

Answer: 192528.0

Prompt: A 955 g sample has specific heat 4.8 J/(g*K). How much heat is needed to raise its temperature by 42 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 955 * 4.8 * 42.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.35 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1324.1466, "unit": "g"}

Answer: 1324.1466

Prompt: A sample contains 7.35 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 = 7.35 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


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": [-5.8, 6.25], "unit": "dimensionless"}

Answer: -5.8

Answer: 6.25

Prompt: A line passes through points (2.5, -8.25) and (8.5, -43.05). 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. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: left_boundary, right_boundary.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["left_boundary", "right_boundary"]}

Answer: INSUFFICIENT_INFO

Prompt: Solve the 1D heat equation after one finite-difference step. The prompt gives r=0.06 and interior values [2.2, 2.1, -1.65], but gives no boundary values. 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: left_boundary, right_boundary.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-0.2)=-3.62 and f(3.7)=22.12.
3. The interval width is 3.9.
Final: {"answer": 36.075, "unit": "area_units"}

Answer: 36.075

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=6.6x+-2.3 from x=-0.2 to x=3.7. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-0.2)=-3.62 and f(3.7)=22.12.
The interval width is 3.9.


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 pairwise_ratios_005200(numerators, denominators):
    if len(numerators) != len(denominators):
        raise ValueError('length mismatch')
    out = []
    for numerator, denominator in zip(numerators, denominators):
        if denominator == 0:
            raise ValueError('zero denominator')
        out.append(numerator / denominator)
    return out
```
Final: {"answer": "python_code", "function": "pairwise_ratios_005200"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_005200(numerators, denominators) that returns a list of n/d values. Raise ValueError if lengths differ or any denominator 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.

try:
    pairwise_ratios_005200([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_005200([1, 2], [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: 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)=34.5 and h=0.95, 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 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 = 3.71 * 8.314 * 598 / 15.5.
Final: {"answer": 1190.0177, "unit": "kPa"}

Answer: 1190.0177

Prompt: An ideal gas sample has n=3.71 mol, T=598 K, and V=15.5 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 = 3.71 * 8.314 * 598 / 15.5.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1415 * 4.31 * 35.
Final: {"answer": 213452.75, "unit": "J"}

Answer: 213452.75

Prompt: A 1415 g sample has specific heat 4.31 J/(g*K). How much heat is needed to raise its temperature by 35 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1415 * 4.31 * 35.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.27 y, so each step multiplies y by 1 - h*0.27.
3. Apply that update for 8 steps and round the final state.
Final: {"answer": 0.0714, "unit": "state_units"}

Answer: 0.0714

Prompt: Use explicit Euler on dy/dt = -0.27 y with y(0)=0.5, step size h=0.8, for 8 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.27 y, so each step multiplies y by 1 - h*0.27.
Apply that update for 8 steps and round the final state.


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_005218(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_005218"}

Answer: python_code

Prompt: Write a Python function finite_differences_005218(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_005218([5]) == []

assert finite_differences_005218([]) == []


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.69 y, so each step multiplies y by 1 - h*0.69.
3. Apply that update for 5 steps and round the final state.
Final: {"answer": -0.0627, "unit": "state_units"}

Answer: -0.0627

Prompt: Use explicit Euler on dy/dt = -0.69 y with y(0)=75.5, step size h=1.8, 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) = -0.69 y, so each step multiplies y by 1 - h*0.69.
Apply that update for 5 steps and round the final state.


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=0.04 and add it to the current value.
Final: {"answer": [1.328, 0.388, 1.19], "unit": "temperature_units"}

Answer: 1.328

Answer: 0.388

Answer: 1.19

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=-0.45; current interior values are [1.5, 0.3, 1.3]. Use r=0.04. 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=0.04 and add it to the current value.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 11.35 kg and a = 14.65 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 166.2775, "unit": "N"}

Answer: 166.2775

Prompt: A cart has mass 11.35 kg and measured acceleration 14.65 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 = 11.35 kg and a = 14.65 m/s^2.
The unit kg*m/s^2 is one newton.


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=0.44 and add it to the current value.
Final: {"answer": [-1.454, 1.422, -0.912], "unit": "temperature_units"}

Answer: -1.454

Answer: 1.422

Answer: -0.912

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=-1.65, right=-0.95; current interior values are [0.9, -1.9, 2.85]. Use r=0.44. 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=0.44 and add it to the current value.


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 * 616 / 17.5.
Final: {"answer": 1062.3297, "unit": "kPa"}

Answer: 1062.3297

Prompt: An ideal gas sample has n=3.63 mol, T=616 K, and V=17.5 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 * 616 / 17.5.


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": [-0.5, 9.5], "unit": "dimensionless"}

Answer: -0.5

Answer: 9.5

Prompt: A line passes through points (-14.25, 16.625) and (-6.75, 12.875). 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. 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 pairwise_ratios_005290(numerators, denominators):
    if len(numerators) != len(denominators):
        raise ValueError('length mismatch')
    out = []
    for numerator, denominator in zip(numerators, denominators):
        if denominator == 0:
            raise ValueError('zero denominator')
        out.append(numerator / denominator)
    return out
```
Final: {"answer": "python_code", "function": "pairwise_ratios_005290"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_005290(numerators, denominators) that returns a list of n/d values. Raise ValueError if lengths differ or any denominator 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.

try:
    pairwise_ratios_005290([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_005290([1, 2], [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: left_boundary, right_boundary.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["left_boundary", "right_boundary"]}

Answer: INSUFFICIENT_INFO

Prompt: Solve the 1D heat equation after one finite-difference step. The prompt gives r=0.08 and interior values [-1.55, -0.1, 0.8], but gives no boundary values. 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: left_boundary, right_boundary.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=227 degrees and keep the unit as newtons.
Final: {"answer": [-42.2839, -45.3439], "unit": "N"}

Answer: -42.2839

Answer: -45.3439

Prompt: A vector has magnitude 62 N at 227 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=227 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 4.15 kg and a = 6.85 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 28.4275, "unit": "N"}

Answer: 28.4275

Prompt: A cart has mass 4.15 kg and measured acceleration 6.85 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 = 4.15 kg and a = 6.85 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 = 28.3^2 and multiply by 0.5 * 12.55.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 5025.5847, "unit": "J"}

Answer: 5025.5847

Prompt: A body of mass 12.55 kg moves at 28.3 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 28.3^2 and multiply by 0.5 * 12.55.
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 = 1820 * 1.915 * 42.
Final: {"answer": 146382.6, "unit": "J"}

Answer: 146382.6

Prompt: A 1820 g sample has specific heat 1.915 J/(g*K). How much heat is needed to raise its temperature by 42 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1820 * 1.915 * 42.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 4.35^2 and multiply by 0.5 * 0.3.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2.8384, "unit": "J"}

Answer: 2.8384

Prompt: A body of mass 0.3 kg moves at 4.35 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 4.35^2 and multiply by 0.5 * 0.3.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. energy divided by length is force, not spring constant
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In student derivation case 69115, check dimensional consistency for k = E / x. The left side has dimensions N/m and the right side has dimensions J/m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
energy divided by length is force, not spring constant
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


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=180 V and R=124.2 ohm.
Final: {"answer": 260.8696, "unit": "W"}

Answer: 260.8696

Prompt: A resistor has resistance 124.2 ohm and voltage 180 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=180 V and R=124.2 ohm.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=160 degrees and keep the unit as newtons.
Final: {"answer": [-409.2361, 148.9498], "unit": "N"}

Answer: -409.2361

Answer: 148.9498

Prompt: A vector has magnitude 435.5 N at 160 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=160 degrees and keep the unit as newtons.


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_005362(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_005362"}

Answer: python_code

Prompt: Write a Python function clamp_005362(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_005362(1, 3, 2)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 15.5^2 and multiply by 0.5 * 24.35.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2925.0438, "unit": "J"}

Answer: 2925.0438

Prompt: A body of mass 24.35 kg moves at 15.5 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 15.5^2 and multiply by 0.5 * 24.35.
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 = 1775 * 0.77 * 93.
Final: {"answer": 127107.75, "unit": "J"}

Answer: 127107.75

Prompt: A 1775 g sample has specific heat 0.77 J/(g*K). How much heat is needed to raise its temperature by 93 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1775 * 0.77 * 93.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 6.32 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1138.5859, "unit": "g"}

Answer: 1138.5859

Prompt: A sample contains 6.32 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 = 6.32 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


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=0.08 and add it to the current value.
Final: {"answer": [0.64, 2.14, 0.256], "unit": "temperature_units"}

Answer: 0.64

Answer: 2.14

Answer: 0.256

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=0.25, right=0.7; current interior values are [0.5, 2.5, 0.0]. Use r=0.08. 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=0.08 and add it to the current value.


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": [-7.1, 19.75], "unit": "dimensionless"}

Answer: -7.1

Answer: 19.75

Prompt: A line passes through points (-1.75, 32.175) and (-0.75, 25.075). 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. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 16^2 and multiply by 0.5 * 22.05.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2822.4, "unit": "J"}

Answer: 2822.4

Prompt: A body of mass 22.05 kg moves at 16 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 16^2 and multiply by 0.5 * 22.05.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.65 y, so each step multiplies y by 1 - h*0.65.
3. Apply that update for 6 steps and round the final state.
Final: {"answer": 6.1953, "unit": "state_units"}

Answer: 6.1953

Prompt: Use explicit Euler on dy/dt = -0.65 y with y(0)=65.5, step size h=0.5, for 6 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.65 y, so each step multiplies y by 1 - h*0.65.
Apply that update for 6 steps and round the final state.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=765 mg, t=68.5 h, and T_half=41 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 240.2819, "unit": "mg"}

Answer: 240.2819

Prompt: A radioactive sample starts with 765 mg. Its half-life is 41 hours. How many mg remain after 68.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=765 mg, t=68.5 h, and T_half=41 h.
The remaining amount has the same mass unit as the initial amount.


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=3.23 mol and T=613 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. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 8.75^2 and multiply by 0.5 * 25.75.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 985.7422, "unit": "J"}

Answer: 985.7422

Prompt: A body of mass 25.75 kg moves at 8.75 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 8.75^2 and multiply by 0.5 * 25.75.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 292.0 / 100.
Final: {"answer": 2.92, "unit": "m"}

Answer: 2.92

Prompt: Convert 292 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 292.0 / 100.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 15.3 kg and a = 3.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 46.665, "unit": "N"}

Answer: 46.665

Prompt: A cart has mass 15.3 kg and measured acceleration 3.05 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 = 15.3 kg and a = 3.05 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 = 2.8^2 and multiply by 0.5 * 26.3.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 103.096, "unit": "J"}

Answer: 103.096

Prompt: A body of mass 26.3 kg moves at 2.8 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 2.8^2 and multiply by 0.5 * 26.3.
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 = 1290 * 0.965 * 23.
Final: {"answer": 28631.55, "unit": "J"}

Answer: 28631.55

Prompt: A 1290 g sample has specific heat 0.965 J/(g*K). How much heat is needed to raise its temperature by 23 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1290 * 0.965 * 23.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(2.1)=20.37 and f(8.8)=25.06.
3. The interval width is 6.7.
Final: {"answer": 152.1905, "unit": "area_units"}

Answer: 152.1905

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=0.7x+18.9 from x=2.1 to x=8.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(2.1)=20.37 and f(8.8)=25.06.
The interval width is 6.7.


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=0.32 and add it to the current value.
Final: {"answer": [0.232, 1.206, 1.762], "unit": "temperature_units"}

Answer: 0.232

Answer: 1.206

Answer: 1.762

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=-0.4, right=1.4; current interior values are [-0.2, 1.35, 2.45]. Use r=0.32. 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=0.32 and add it to the current value.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=362.5 mg, t=207.5 h, and T_half=105.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 92.7331, "unit": "mg"}

Answer: 92.7331

Prompt: A radioactive sample starts with 362.5 mg. Its half-life is 105.5 hours. How many mg remain after 207.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=362.5 mg, t=207.5 h, and T_half=105.5 h.
The remaining amount has the same mass unit as the initial amount.


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=0.26 and add it to the current value.
Final: {"answer": [0.593, 0.26, 0.944], "unit": "temperature_units"}

Answer: 0.593

Answer: 0.26

Answer: 0.944

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=2.25; current interior values are [1.1, -0.65, 1.1]. Use r=0.26. 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=0.26 and add it to the current value.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 90 * 4.04 * 60.
Final: {"answer": 21816.0, "unit": "J"}

Answer: 21816.0

Prompt: A 90 g sample has specific heat 4.04 J/(g*K). How much heat is needed to raise its temperature by 60 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 90 * 4.04 * 60.


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

Answer: INSUFFICIENT_INFO

Prompt: Compute projectile range on level ground. The prompt gives initial speed 13.9 m/s but does not give launch angle or vertical velocity. 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: launch_angle_or_vertical_velocity.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=72 degrees through the sine of 2 theta.
3. Substitute v=29.9 m/s and g=9.81 m/s^2.
Final: {"answer": 53.5663, "unit": "m"}

Answer: 53.5663

Prompt: A projectile is launched on level ground with no air resistance at speed 29.9 m/s and angle 72 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=72 degrees through the sine of 2 theta.
Substitute v=29.9 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 = 940 * 2.41 * 10.
Final: {"answer": 22654.0, "unit": "J"}

Answer: 22654.0

Prompt: A 940 g sample has specific heat 2.41 J/(g*K). How much heat is needed to raise its temperature by 10 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 940 * 2.41 * 10.