Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=604 mg, t=201.5 h, and T_half=106.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 162.7356, "unit": "mg"}

Answer: 162.7356

Prompt: A radioactive sample starts with 604 mg. Its half-life is 106.5 hours. How many mg remain after 201.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=604 mg, t=201.5 h, and T_half=106.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.97 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 56.6868, "unit": "g"}

Answer: 56.6868

Prompt: A sample contains 0.97 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 = 0.97 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4860 mL to 4.86 L.
3. Compute M = 2.055 / 4.86.
Final: {"answer": 0.4228, "unit": "mol/L"}

Answer: 0.4228

Prompt: A solution contains 2.055 mol of solute in 4860 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4860 mL to 4.86 L.
Compute M = 2.055 / 4.86.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.14 y, so each step multiplies y by 1 - h*0.14.
3. Apply that update for 5 steps and round the final state.
Final: {"answer": 22.3705, "unit": "state_units"}

Answer: 22.3705

Prompt: Use explicit Euler on dy/dt = -0.14 y with y(0)=79.5, step size h=1.6, 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) = -0.14 y, so each step multiplies y by 1 - h*0.14.
Apply that update for 5 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.5)=-6.05 and f(0.9)=-5.51.
3. The interval width is 5.4.
Final: {"answer": -31.212, "unit": "area_units"}

Answer: -31.212

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=0.1x+-5.6 from x=-4.5 to x=0.9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.5)=-6.05 and f(0.9)=-5.51.
The interval width is 5.4.

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


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 17717.0 / 1000.
Final: {"answer": 17.717, "unit": "kg"}

Answer: 17.717

Prompt: Convert 17717 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 17717.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=310 degrees and keep the unit as newtons.
Final: {"answer": [49.816, -59.3684], "unit": "N"}

Answer: 49.816

Answer: -59.3684

Prompt: A vector has magnitude 77.5 N at 310 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=310 degrees and keep the unit as newtons.

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


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-6.75, -2.25], "unit": "dimensionless"}

Answer: -6.75

Answer: -2.25

Prompt: A line passes through points (-4, 24.75) and (-1, 4.5). 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. density times acceleration times height gives kg/(m*s^2), or pascal
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In instrument manual case 67778, check dimensional consistency for p = rho * g * h. The left side has dimensions Pa and the right side has dimensions Pa. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
density times acceleration times height gives kg/(m*s^2), or pascal
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 pairwise_ratios_006712(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_006712"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_006712(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_006712([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_006712([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=0.1 and interior values [0.4, 0.85, 1.6], 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. Use Newton's second law F = m a.
2. Substitute m = 19.65 kg and a = 15.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 300.645, "unit": "N"}

Answer: 300.645

Prompt: A cart has mass 19.65 kg and measured acceleration 15.3 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 = 19.65 kg and a = 15.3 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=35 degrees through the sine of 2 theta.
3. Substitute v=50.3 m/s and g=9.81 m/s^2.
Final: {"answer": 242.3554, "unit": "m"}

Answer: 242.3554

Prompt: A projectile is launched on level ground with no air resistance at speed 50.3 m/s and angle 35 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=35 degrees through the sine of 2 theta.
Substitute v=50.3 m/s and g=9.81 m/s^2.

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


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.83 * 8.314 * 412 / 24.4.
Final: {"answer": 818.4383, "unit": "kPa"}

Answer: 818.4383

Prompt: An ideal gas sample has n=5.83 mol, T=412 K, and V=24.4 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 5.83 * 8.314 * 412 / 24.4.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 740 * 2.095 * 56.
Final: {"answer": 86816.8, "unit": "J"}

Answer: 86816.8

Prompt: A 740 g sample has specific heat 2.095 J/(g*K). How much heat is needed to raise its temperature by 56 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 740 * 2.095 * 56.


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=222.3 V and R=8.3 ohm.
Final: {"answer": 5953.8904, "unit": "W"}

Answer: 5953.8904

Prompt: A resistor has resistance 8.3 ohm and voltage 222.3 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=222.3 V and R=8.3 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=837.5 mg, t=26.5 h, and T_half=108.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 707.0683, "unit": "mg"}

Answer: 707.0683

Prompt: A radioactive sample starts with 837.5 mg. Its half-life is 108.5 hours. How many mg remain after 26.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=837.5 mg, t=26.5 h, and T_half=108.5 h.
The remaining amount has the same mass unit as the initial amount.

initial_amount * 0.5 ** (elapsed_time / half_life)


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.37 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 6.6656, "unit": "g"}

Answer: 6.6656

Prompt: A sample contains 0.37 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.37 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4435 mL to 4.435 L.
3. Compute M = 3.51 / 4.435.
Final: {"answer": 0.7914, "unit": "mol/L"}

Answer: 0.7914

Prompt: A solution contains 3.51 mol of solute in 4435 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4435 mL to 4.435 L.
Compute M = 3.51 / 4.435.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(2)=13.3 and f(7.1)=22.48.
3. The interval width is 5.1.
Final: {"answer": 91.239, "unit": "area_units"}

Answer: 91.239

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=1.8x+9.7 from x=2 to x=7.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(2)=13.3 and f(7.1)=22.48.
The interval width is 5.1.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.44 and add it to the current value.
Final: {"answer": [0.768, 0.308, 1.472], "unit": "temperature_units"}

Answer: 0.768

Answer: 0.308

Answer: 1.472

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=1.95, right=2.95; current interior values are [-0.75, 0.0, 1.45]. 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. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 3790.0 / 1000.
Final: {"answer": 3.79, "unit": "kg"}

Answer: 3.79

Prompt: Convert 3790 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 3790.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=15 degrees and keep the unit as newtons.
Final: {"answer": [370.9155, 99.3865], "unit": "N"}

Answer: 370.9155

Answer: 99.3865

Prompt: A vector has magnitude 384 N at 15 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=15 degrees and keep the unit as newtons.


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 instrument manual case 437754, 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. 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 weighted_sum_006730(values, weights):
    if len(values) != len(weights):
        raise ValueError('length mismatch')
    return sum(value * weight for value, weight in zip(values, weights))
```
Final: {"answer": "python_code", "function": "weighted_sum_006730"}

Answer: python_code

Prompt: Write a Python function weighted_sum_006730(values, weights) that returns sum(value*weight). Raise ValueError when the two inputs have different lengths. 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:
    weighted_sum_006730([1], [1, 2])
    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)=18 and h=1.9, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 10.6 kg and a = 12.9 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 136.74, "unit": "N"}

Answer: 136.74

Prompt: A cart has mass 10.6 kg and measured acceleration 12.9 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 10.6 kg and a = 12.9 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 = 5.65^2 and multiply by 0.5 * 22.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 355.1378, "unit": "J"}

Answer: 355.1378

Prompt: A body of mass 22.25 kg moves at 5.65 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 5.65^2 and multiply by 0.5 * 22.25.
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=58 degrees through the sine of 2 theta.
3. Substitute v=83.1 m/s and g=9.81 m/s^2.
Final: {"answer": 632.6933, "unit": "m"}

Answer: 632.6933

Prompt: A projectile is launched on level ground with no air resistance at speed 83.1 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=83.1 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.34 * 8.314 * 530 / 99.6.
Final: {"answer": 147.7655, "unit": "kPa"}

Answer: 147.7655

Prompt: An ideal gas sample has n=3.34 mol, T=530 K, and V=99.6 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 3.34 * 8.314 * 530 / 99.6.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1425 * 4.93 * 31.
Final: {"answer": 217782.75, "unit": "J"}

Answer: 217782.75

Prompt: A 1425 g sample has specific heat 4.93 J/(g*K). How much heat is needed to raise its temperature by 31 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1425 * 4.93 * 31.


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=182.9 V and R=36.8 ohm.
Final: {"answer": 909.0329, "unit": "W"}

Answer: 909.0329

Prompt: A resistor has resistance 36.8 ohm and voltage 182.9 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=182.9 V and R=36.8 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.51 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 330.5151, "unit": "g"}

Answer: 330.5151

Prompt: A sample contains 7.51 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 = 7.51 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3350 mL to 3.35 L.
3. Compute M = 3.48 / 3.35.
Final: {"answer": 1.0388, "unit": "mol/L"}

Answer: 1.0388

Prompt: A solution contains 3.48 mol of solute in 3350 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3350 mL to 3.35 L.
Compute M = 3.48 / 3.35.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.94 y, so each step multiplies y by 1 - h*0.94.
3. Apply that update for 8 steps and round the final state.
Final: {"answer": 0.0529, "unit": "state_units"}

Answer: 0.0529

Prompt: Use explicit Euler on dy/dt = -0.94 y with y(0)=40.5, step size h=0.6, 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) = -0.94 y, so each step multiplies y by 1 - h*0.94.
Apply that update for 8 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=0.32 and add it to the current value.
Final: {"answer": [1.87, 1.75, 1.978], "unit": "temperature_units"}

Answer: 1.87

Answer: 1.75

Answer: 1.978

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=1.55, right=2.45; current interior values are [1.55, 2.55, 1.05]. 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. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 43213.0 / 1000.
Final: {"answer": 43.213, "unit": "kg"}

Answer: 43.213

Prompt: Convert 43213 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 43213.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=19 degrees and keep the unit as newtons.
Final: {"answer": [401.3726, 138.2037], "unit": "N"}

Answer: 401.3726

Answer: 138.2037

Prompt: A vector has magnitude 424.5 N at 19 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=19 degrees and keep the unit as newtons.


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

Answer: consistent

Prompt: In student derivation case 731468, check dimensional consistency for R = v^2 * sin(theta) / g. The left side has dimensions m and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

```python
def center_values_006748(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_006748"}

Answer: python_code

Prompt: Write a Python function center_values_006748(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_006748([])
    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 allocation scoring problem with 280 items and budget 60470, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 16.95 kg and a = 1.45 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 24.5775, "unit": "N"}

Answer: 24.5775

Prompt: A cart has mass 16.95 kg and measured acceleration 1.45 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 16.95 kg and a = 1.45 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=11 degrees through the sine of 2 theta.
3. Substitute v=80 m/s and g=9.81 m/s^2.
Final: {"answer": 244.3917, "unit": "m"}

Answer: 244.3917

Prompt: A projectile is launched on level ground with no air resistance at speed 80 m/s and angle 11 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=11 degrees through the sine of 2 theta.
Substitute v=80 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 = 375 * 1.645 * 39.
Final: {"answer": 24058.125, "unit": "J"}

Answer: 24058.125

Prompt: A 375 g sample has specific heat 1.645 J/(g*K). How much heat is needed to raise its temperature by 39 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 375 * 1.645 * 39.


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=40.5 V and R=189.5 ohm.
Final: {"answer": 8.6557, "unit": "W"}

Answer: 8.6557

Prompt: A resistor has resistance 189.5 ohm and voltage 40.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=40.5 V and R=189.5 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=885 mg, t=295.5 h, and T_half=17 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0052, "unit": "mg"}

Answer: 0.0052

Prompt: A radioactive sample starts with 885 mg. Its half-life is 17 hours. How many mg remain after 295.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=885 mg, t=295.5 h, and T_half=17 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 2560 mL to 2.56 L.
3. Compute M = 3.405 / 2.56.
Final: {"answer": 1.3301, "unit": "mol/L"}

Answer: 1.3301

Prompt: A solution contains 3.405 mol of solute in 2560 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2560 mL to 2.56 L.
Compute M = 3.405 / 2.56.


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 10 steps and round the final state.
Final: {"answer": 0.0, "unit": "state_units"}

Answer: 0.0

Prompt: Use explicit Euler on dy/dt = -0.56 y with y(0)=49, 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) = -0.56 y, so each step multiplies y by 1 - h*0.56.
Apply that update for 10 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=0.06 and add it to the current value.
Final: {"answer": [0.176, 0.615, 1.091], "unit": "temperature_units"}

Answer: 0.176

Answer: 0.615

Answer: 1.091

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=-0.6, right=-0.75; current interior values are [0.2, 0.6, 1.25]. Use r=0.06. 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=0.06 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=281 degrees and keep the unit as newtons.
Final: {"answer": [75.465, -388.2336], "unit": "N"}

Answer: 75.465

Answer: -388.2336

Prompt: A vector has magnitude 395.5 N at 281 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=281 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": [-6.1, 0.75], "unit": "dimensionless"}

Answer: -6.1

Answer: 0.75

Prompt: A line passes through points (-15, 92.25) and (-8.25, 51.075). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: consistent

Prompt: In field worksheet case 79857, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

```python
def safe_mean_006766(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_006766"}

Answer: python_code

Prompt: Write a Python function safe_mean_006766(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_006766([])
    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=0.46 and interior values [0.5, -2.0, 1.1], 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. Use Newton's second law F = m a.
2. Substitute m = 15.8 kg and a = 9.45 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 149.31, "unit": "N"}

Answer: 149.31

Prompt: A cart has mass 15.8 kg and measured acceleration 9.45 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 15.8 kg and a = 9.45 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=54 degrees through the sine of 2 theta.
3. Substitute v=59.6 m/s and g=9.81 m/s^2.
Final: {"answer": 344.3736, "unit": "m"}

Answer: 344.3736

Prompt: A projectile is launched on level ground with no air resistance at speed 59.6 m/s and angle 54 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=54 degrees through the sine of 2 theta.
Substitute v=59.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.09 * 8.314 * 370 / 31.6.
Final: {"answer": 495.4986, "unit": "kPa"}

Answer: 495.4986

Prompt: An ideal gas sample has n=5.09 mol, T=370 K, and V=31.6 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 5.09 * 8.314 * 370 / 31.6.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 840 * 3.97 * 106.
Final: {"answer": 353488.8, "unit": "J"}

Answer: 353488.8

Prompt: A 840 g sample has specific heat 3.97 J/(g*K). How much heat is needed to raise its temperature by 106 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 840 * 3.97 * 106.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=22.9 V and R=59 ohm.
Final: {"answer": 8.8883, "unit": "W"}

Answer: 8.8883

Prompt: A resistor has resistance 59 ohm and voltage 22.9 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=22.9 V and R=59 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=18 mg, t=72.5 h, and T_half=76 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 9.2919, "unit": "mg"}

Answer: 9.2919

Prompt: A radioactive sample starts with 18 mg. Its half-life is 76 hours. How many mg remain after 72.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=18 mg, t=72.5 h, and T_half=76 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 2940 mL to 2.94 L.
3. Compute M = 4.98 / 2.94.
Final: {"answer": 1.6939, "unit": "mol/L"}

Answer: 1.6939

Prompt: A solution contains 4.98 mol of solute in 2940 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2940 mL to 2.94 L.
Compute M = 4.98 / 2.94.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.32 y, so each step multiplies y by 1 - h*0.32.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 0.0035, "unit": "state_units"}

Answer: 0.0035

Prompt: Use explicit Euler on dy/dt = -0.32 y with y(0)=68, step size h=1.85, 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) = -0.32 y, so each step multiplies y by 1 - h*0.32.
Apply that update for 11 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(0.5)=10.4 and f(7.5)=-33.
3. The interval width is 7.
Final: {"answer": -79.1, "unit": "area_units"}

Answer: -79.1

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-6.2x+13.5 from x=0.5 to x=7.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(0.5)=10.4 and f(7.5)=-33.
The interval width is 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=0.38 and add it to the current value.
Final: {"answer": [0.524, 1.18, -0.421], "unit": "temperature_units"}

Answer: 0.524

Answer: 1.18

Answer: -0.421

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=-2; current interior values are [1.55, -0.15, 1.65]. 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=272 degrees and keep the unit as newtons.
Final: {"answer": [1.8846, -53.9671], "unit": "N"}

Answer: 1.8846

Answer: -53.9671

Prompt: A vector has magnitude 54 N at 272 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=272 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": [4.35, 0.5], "unit": "dimensionless"}

Answer: 4.35

Answer: 0.5

Prompt: A line passes through points (4, 17.9) and (7.5, 33.125). 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. velocity divided by time is acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In simulation spec case 895028, check dimensional consistency for x = v / t. The left side has dimensions m and the right side has dimensions m/s^2. Return consistent or inconsistent with a brief reason.

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

Answer: inconsistent


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

```python
def normalize_006784(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_006784"}

Answer: python_code

Prompt: Write a Python function normalize_006784(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_006784([2, 3, 5]); assert len(out) == 3

try:
    normalize_006784([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: 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=0.22 and interior values [2.2, 0.25, 1.4], 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. Use Newton's second law F = m a.
2. Substitute m = 6.3 kg and a = 13.2 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 83.16, "unit": "N"}

Answer: 83.16

Prompt: A cart has mass 6.3 kg and measured acceleration 13.2 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 6.3 kg and a = 13.2 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 23.75^2 and multiply by 0.5 * 8.9.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2510.0781, "unit": "J"}

Answer: 2510.0781

Prompt: A body of mass 8.9 kg moves at 23.75 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 23.75^2 and multiply by 0.5 * 8.9.
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=38 degrees through the sine of 2 theta.
3. Substitute v=26.5 m/s and g=9.81 m/s^2.
Final: {"answer": 69.4587, "unit": "m"}

Answer: 69.4587

Prompt: A projectile is launched on level ground with no air resistance at speed 26.5 m/s and angle 38 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=38 degrees through the sine of 2 theta.
Substitute v=26.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.93 * 8.314 * 460 / 111.9.
Final: {"answer": 134.3168, "unit": "kPa"}

Answer: 134.3168

Prompt: An ideal gas sample has n=3.93 mol, T=460 K, and V=111.9 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 = 3.93 * 8.314 * 460 / 111.9.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1570 * 3.065 * 48.
Final: {"answer": 230978.4, "unit": "J"}

Answer: 230978.4

Prompt: A 1570 g sample has specific heat 3.065 J/(g*K). How much heat is needed to raise its temperature by 48 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1570 * 3.065 * 48.


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=66.5 V and R=199.9 ohm.
Final: {"answer": 22.1223, "unit": "W"}

Answer: 22.1223

Prompt: A resistor has resistance 199.9 ohm and voltage 66.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=66.5 V and R=199.9 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=207 mg, t=71 h, and T_half=49 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 75.82, "unit": "mg"}

Answer: 75.82

Prompt: A radioactive sample starts with 207 mg. Its half-life is 49 hours. How many mg remain after 71 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=207 mg, t=71 h, and T_half=49 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 4.06 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 178.6806, "unit": "g"}

Answer: 178.6806

Prompt: A sample contains 4.06 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 4.06 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 420 mL to 0.42 L.
3. Compute M = 0.875 / 0.42.
Final: {"answer": 2.0833, "unit": "mol/L"}

Answer: 2.0833

Prompt: A solution contains 0.875 mol of solute in 420 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 420 mL to 0.42 L.
Compute M = 0.875 / 0.42.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(9.1)=-62.58 and f(17)=-124.2.
3. The interval width is 7.9.
Final: {"answer": -737.781, "unit": "area_units"}

Answer: -737.781

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-7.8x+8.4 from x=9.1 to x=17. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(9.1)=-62.58 and f(17)=-124.2.
The interval width is 7.9.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 4204.0 / 100.
Final: {"answer": 42.04, "unit": "m"}

Answer: 42.04

Prompt: Convert 4204 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 4204.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=278 degrees and keep the unit as newtons.
Final: {"answer": [26.93, -191.6169], "unit": "N"}

Answer: 26.93

Answer: -191.6169

Prompt: A vector has magnitude 193.5 N at 278 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=278 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": [5.6, -12.75], "unit": "dimensionless"}

Answer: 5.6

Answer: -12.75

Prompt: A line passes through points (9.75, 41.85) and (14, 65.65). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. 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 133967, 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. 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_006802(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_006802"}

Answer: python_code

Prompt: Write a Python function unit_vector_006802(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_006802(0, 0)
    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: required_fields, allowed_transformations.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["required_fields", "allowed_transformations"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a sanitizer for valid warehouse survey records in release 812789, but the prompt does not state which fields are required or which transformations are allowed. 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: required_fields, allowed_transformations.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 18.8 kg and a = 12.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 231.24, "unit": "N"}

Answer: 231.24

Prompt: A cart has mass 18.8 kg and measured acceleration 12.3 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 = 18.8 kg and a = 12.3 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 30.3^2 and multiply by 0.5 * 26.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 12049.9312, "unit": "J"}

Answer: 12049.9312

Prompt: A body of mass 26.25 kg moves at 30.3 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 30.3^2 and multiply by 0.5 * 26.25.
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=39 degrees through the sine of 2 theta.
3. Substitute v=26.6 m/s and g=9.81 m/s^2.
Final: {"answer": 70.5503, "unit": "m"}

Answer: 70.5503

Prompt: A projectile is launched on level ground with no air resistance at speed 26.6 m/s and angle 39 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=39 degrees through the sine of 2 theta.
Substitute v=26.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.12 * 8.314 * 564 / 82.6.
Final: {"answer": 177.1184, "unit": "kPa"}

Answer: 177.1184

Prompt: An ideal gas sample has n=3.12 mol, T=564 K, and V=82.6 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 3.12 * 8.314 * 564 / 82.6.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 845 * 3.065 * 83.
Final: {"answer": 214963.775, "unit": "J"}

Answer: 214963.775

Prompt: A 845 g sample has specific heat 3.065 J/(g*K). How much heat is needed to raise its temperature by 83 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 845 * 3.065 * 83.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=219 mg, t=303 h, and T_half=75.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 13.5624, "unit": "mg"}

Answer: 13.5624

Prompt: A radioactive sample starts with 219 mg. Its half-life is 75.5 hours. How many mg remain after 303 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=219 mg, t=303 h, and T_half=75.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.84 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 165.9696, "unit": "g"}

Answer: 165.9696

Prompt: A sample contains 2.84 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 = 2.84 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3800 mL to 3.8 L.
3. Compute M = 3.82 / 3.8.
Final: {"answer": 1.0053, "unit": "mol/L"}

Answer: 1.0053

Prompt: A solution contains 3.82 mol of solute in 3800 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3800 mL to 3.8 L.
Compute M = 3.82 / 3.8.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.71 y, so each step multiplies y by 1 - h*0.71.
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 = -0.71 y with y(0)=75.5, step size h=1.35, 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) = -0.71 y, so each step multiplies y by 1 - h*0.71.
Apply that update for 10 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(2.6)=8.06 and f(3.8)=3.98.
3. The interval width is 1.2.
Final: {"answer": 7.224, "unit": "area_units"}

Answer: 7.224

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.4x+16.9 from x=2.6 to x=3.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(2.6)=8.06 and f(3.8)=3.98.
The interval width is 1.2.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 2061.0 / 1000.
Final: {"answer": 2.061, "unit": "kg"}

Answer: 2.061

Prompt: Convert 2061 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 2061.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=356 degrees and keep the unit as newtons.
Final: {"answer": [197.0189, -13.7769], "unit": "N"}

Answer: 197.0189

Answer: -13.7769

Prompt: A vector has magnitude 197.5 N at 356 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=356 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=0.12 and interior values [-1.9, 2.55, 2.75], 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. Use Newton's second law F = m a.
2. Substitute m = 16 kg and a = 13.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 208.8, "unit": "N"}

Answer: 208.8

Prompt: A cart has mass 16 kg and measured acceleration 13.05 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 = 16 kg and a = 13.05 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 = 10.25^2 and multiply by 0.5 * 8.85.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 464.9016, "unit": "J"}

Answer: 464.9016

Prompt: A body of mass 8.85 kg moves at 10.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 10.25^2 and multiply by 0.5 * 8.85.
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=28 degrees through the sine of 2 theta.
3. Substitute v=30.9 m/s and g=9.81 m/s^2.
Final: {"answer": 80.6905, "unit": "m"}

Answer: 80.6905

Prompt: A projectile is launched on level ground with no air resistance at speed 30.9 m/s and angle 28 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=28 degrees through the sine of 2 theta.
Substitute v=30.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 2.51 * 8.314 * 609 / 111.5.
Final: {"answer": 113.9793, "unit": "kPa"}

Answer: 113.9793

Prompt: An ideal gas sample has n=2.51 mol, T=609 K, and V=111.5 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 = 2.51 * 8.314 * 609 / 111.5.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 545 * 3.66 * 103.
Final: {"answer": 205454.1, "unit": "J"}

Answer: 205454.1

Prompt: A 545 g sample has specific heat 3.66 J/(g*K). How much heat is needed to raise its temperature by 103 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 545 * 3.66 * 103.


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=110.5 V and R=101.2 ohm.
Final: {"answer": 120.6546, "unit": "W"}

Answer: 120.6546

Prompt: A resistor has resistance 101.2 ohm and voltage 110.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=110.5 V and R=101.2 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 6.73 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 121.241, "unit": "g"}

Answer: 121.241

Prompt: A sample contains 6.73 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.73 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 520 mL to 0.52 L.
3. Compute M = 3.245 / 0.52.
Final: {"answer": 6.2404, "unit": "mol/L"}

Answer: 6.2404

Prompt: A solution contains 3.245 mol of solute in 520 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 520 mL to 0.52 L.
Compute M = 3.245 / 0.52.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.17 y, so each step multiplies y by 1 - h*0.17.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 21.7293, "unit": "state_units"}

Answer: 21.7293

Prompt: Use explicit Euler on dy/dt = -0.17 y with y(0)=31, 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) = -0.17 y, so each step multiplies y by 1 - h*0.17.
Apply that update for 4 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.1)=-12.18 and f(3.2)=44.76.
3. The interval width is 7.3.
Final: {"answer": 118.917, "unit": "area_units"}

Answer: 118.917

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=7.8x+19.8 from x=-4.1 to x=3.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.1)=-12.18 and f(3.2)=44.76.
The interval width is 7.3.


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": [1.775, 1.665, 0.83], "unit": "temperature_units"}

Answer: 1.775

Answer: 1.665

Answer: 0.83

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=0.2, right=0.45; current interior values are [2.15, 2.85, -0.4]. 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 71.0 + 273.15.
Final: {"answer": 344.15, "unit": "K"}

Answer: 344.15

Prompt: Convert 71 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 71.0 + 273.15.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=344 degrees and keep the unit as newtons.
Final: {"answer": [162.4532, -46.5827], "unit": "N"}

Answer: 162.4532

Answer: -46.5827

Prompt: A vector has magnitude 169 N at 344 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=344 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. coulomb per second is ampere
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In student derivation case 236577, check dimensional consistency for I = Q / t. The left side has dimensions A and the right side has dimensions A. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
coulomb per second is ampere
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 clamp_006838(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_006838"}

Answer: python_code

Prompt: Write a Python function clamp_006838(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_006838(1, 3, 2)
    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 allocation scoring problem with 55 items and budget 71985, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 3.95 kg and a = 2.8 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 11.06, "unit": "N"}

Answer: 11.06

Prompt: A cart has mass 3.95 kg and measured acceleration 2.8 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 3.95 kg and a = 2.8 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 31.1^2 and multiply by 0.5 * 16.1.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 7786.0405, "unit": "J"}

Answer: 7786.0405

Prompt: A body of mass 16.1 kg moves at 31.1 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 31.1^2 and multiply by 0.5 * 16.1.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=6 degrees through the sine of 2 theta.
3. Substitute v=66.5 m/s and g=9.81 m/s^2.
Final: {"answer": 93.7245, "unit": "m"}

Answer: 93.7245

Prompt: A projectile is launched on level ground with no air resistance at speed 66.5 m/s and angle 6 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=6 degrees through the sine of 2 theta.
Substitute v=66.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.05 * 8.314 * 567 / 118.5.
Final: {"answer": 121.3318, "unit": "kPa"}

Answer: 121.3318

Prompt: An ideal gas sample has n=3.05 mol, T=567 K, and V=118.5 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 = 3.05 * 8.314 * 567 / 118.5.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1955 * 1.085 * 21.
Final: {"answer": 44544.675, "unit": "J"}

Answer: 44544.675

Prompt: A 1955 g sample has specific heat 1.085 J/(g*K). How much heat is needed to raise its temperature by 21 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1955 * 1.085 * 21.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=931 mg, t=219.5 h, and T_half=7 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0, "unit": "mg"}

Answer: 0.0

Prompt: A radioactive sample starts with 931 mg. Its half-life is 7 hours. How many mg remain after 219.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=931 mg, t=219.5 h, and T_half=7 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.59 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 25.9659, "unit": "g"}

Answer: 25.9659

Prompt: A sample contains 0.59 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 = 0.59 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 820 mL to 0.82 L.
3. Compute M = 2.625 / 0.82.
Final: {"answer": 3.2012, "unit": "mol/L"}

Answer: 3.2012

Prompt: A solution contains 2.625 mol of solute in 820 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 820 mL to 0.82 L.
Compute M = 2.625 / 0.82.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.62 y, so each step multiplies y by 1 - h*0.62.
3. Apply that update for 2 steps and round the final state.
Final: {"answer": 0.2926, "unit": "state_units"}

Answer: 0.2926

Prompt: Use explicit Euler on dy/dt = -0.62 y with y(0)=40.5, step size h=1.75, for 2 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.62 y, so each step multiplies y by 1 - h*0.62.
Apply that update for 2 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(5.2)=34.72 and f(6.5)=42.65.
3. The interval width is 1.3.
Final: {"answer": 50.2905, "unit": "area_units"}

Answer: 50.2905

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=6.1x+3 from x=5.2 to x=6.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(5.2)=34.72 and f(6.5)=42.65.
The interval width is 1.3.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 347.0 / 100.
Final: {"answer": 3.47, "unit": "m"}

Answer: 3.47

Prompt: Convert 347 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 347.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=59 degrees and keep the unit as newtons.
Final: {"answer": [134.1674, 223.2921], "unit": "N"}

Answer: 134.1674

Answer: 223.2921

Prompt: A vector has magnitude 260.5 N at 59 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=59 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": [-7.85, -15.75], "unit": "dimensionless"}

Answer: -7.85

Answer: -15.75

Prompt: A line passes through points (-3.25, 9.7625) and (8.75, -84.4375). 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. velocity times time gives distance, not acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In instrument manual case 29471, check dimensional consistency for a = v * t. The left side has dimensions m/s^2 and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity times time gives distance, not acceleration
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 = 1.9 kg and a = 10.15 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 19.285, "unit": "N"}

Answer: 19.285

Prompt: A cart has mass 1.9 kg and measured acceleration 10.15 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 = 10.15 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 = 41.65^2 and multiply by 0.5 * 8.4.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 7285.8345, "unit": "J"}

Answer: 7285.8345

Prompt: A body of mass 8.4 kg moves at 41.65 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 41.65^2 and multiply by 0.5 * 8.4.
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 = 0.76 * 8.314 * 352 / 28.1.
Final: {"answer": 79.1516, "unit": "kPa"}

Answer: 79.1516

Prompt: An ideal gas sample has n=0.76 mol, T=352 K, and V=28.1 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 = 0.76 * 8.314 * 352 / 28.1.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=58.7 V and R=107.7 ohm.
Final: {"answer": 31.9934, "unit": "W"}

Answer: 31.9934

Prompt: A resistor has resistance 107.7 ohm and voltage 58.7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=58.7 V and R=107.7 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=865 mg, t=70.5 h, and T_half=75 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 450.8664, "unit": "mg"}

Answer: 450.8664

Prompt: A radioactive sample starts with 865 mg. Its half-life is 75 hours. How many mg remain after 70.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=865 mg, t=70.5 h, and T_half=75 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 4740 mL to 4.74 L.
3. Compute M = 3.69 / 4.74.
Final: {"answer": 0.7785, "unit": "mol/L"}

Answer: 0.7785

Prompt: A solution contains 3.69 mol of solute in 4740 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4740 mL to 4.74 L.
Compute M = 3.69 / 4.74.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.43 y, so each step multiplies y by 1 - h*0.43.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": 5.2366, "unit": "state_units"}

Answer: 5.2366

Prompt: Use explicit Euler on dy/dt = -0.43 y with y(0)=20.5, step size h=0.85, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.43 y, so each step multiplies y by 1 - h*0.43.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.8)=11.2 and f(-4.1)=7.
3. The interval width is 0.7.
Final: {"answer": 6.37, "unit": "area_units"}

Answer: 6.37

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-6x+-17.6 from x=-4.8 to x=-4.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.8)=11.2 and f(-4.1)=7.
The interval width is 0.7.


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=0.04 and add it to the current value.
Final: {"answer": [-0.392, 0.248, 0.852], "unit": "temperature_units"}

Answer: -0.392

Answer: 0.248

Answer: 0.852

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=1.45, right=-0.8; current interior values are [-0.5, 0.25, 0.95]. 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. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 6540.0 / 1000.
Final: {"answer": 6.54, "unit": "kg"}

Answer: 6.54

Prompt: Convert 6540 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 6540.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=79 degrees and keep the unit as newtons.
Final: {"answer": [26.9995, 138.9002], "unit": "N"}

Answer: 26.9995

Answer: 138.9002

Prompt: A vector has magnitude 141.5 N at 79 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=79 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [0.05, -26.0], "unit": "dimensionless"}

Answer: 0.05

Answer: -26.0

Prompt: A line passes through points (18.75, -25.0625) and (19, -25.05). 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. 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 lab notebook case 864168, 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. 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=0.28 and interior values [-0.85, -1.45, 0.75], 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. Use Newton's second law F = m a.
2. Substitute m = 21.35 kg and a = 2.2 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 46.97, "unit": "N"}

Answer: 46.97

Prompt: A cart has mass 21.35 kg and measured acceleration 2.2 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 = 21.35 kg and a = 2.2 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 = 30.95^2 and multiply by 0.5 * 12.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 5747.415, "unit": "J"}

Answer: 5747.415

Prompt: A body of mass 12 kg moves at 30.95 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 30.95^2 and multiply by 0.5 * 12.
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 = 1355 * 4.165 * 28.
Final: {"answer": 158020.1, "unit": "J"}

Answer: 158020.1

Prompt: A 1355 g sample has specific heat 4.165 J/(g*K). How much heat is needed to raise its temperature by 28 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1355 * 4.165 * 28.


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=140 V and R=140.2 ohm.
Final: {"answer": 139.8003, "unit": "W"}

Answer: 139.8003

Prompt: A resistor has resistance 140.2 ohm and voltage 140 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=140 V and R=140.2 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=961.5 mg, t=43.5 h, and T_half=12 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 77.932, "unit": "mg"}

Answer: 77.932

Prompt: A radioactive sample starts with 961.5 mg. Its half-life is 12 hours. How many mg remain after 43.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=961.5 mg, t=43.5 h, and T_half=12 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.64 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 47.5596, "unit": "g"}

Answer: 47.5596

Prompt: A sample contains 2.64 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 2.64 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 1335 mL to 1.335 L.
3. Compute M = 2.94 / 1.335.
Final: {"answer": 2.2022, "unit": "mol/L"}

Answer: 2.2022

Prompt: A solution contains 2.94 mol of solute in 1335 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1335 mL to 1.335 L.
Compute M = 2.94 / 1.335.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.17 y, so each step multiplies y by 1 - h*0.17.
3. Apply that update for 5 steps and round the final state.
Final: {"answer": 4.8289, "unit": "state_units"}

Answer: 4.8289

Prompt: Use explicit Euler on dy/dt = -0.17 y with y(0)=6, step size h=0.25, for 5 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.17 y, so each step multiplies y by 1 - h*0.17.
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=0.32 and add it to the current value.
Final: {"answer": [0.248, -0.424, -1.438], "unit": "temperature_units"}

Answer: 0.248

Answer: -0.424

Answer: -1.438

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=0.15, right=-1.35; current interior values are [1.8, -1.4, -1.55]. 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. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 726.0 / 100.
Final: {"answer": 7.26, "unit": "m"}

Answer: 7.26

Prompt: Convert 726 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 726.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=88 degrees and keep the unit as newtons.
Final: {"answer": [10.086, 288.8239], "unit": "N"}

Answer: 10.086

Answer: 288.8239

Prompt: A vector has magnitude 289 N at 88 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=88 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-2.1, -6.5], "unit": "dimensionless"}

Answer: -2.1

Answer: -6.5

Prompt: A line passes through points (-10, 14.5) and (1, -8.6). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity times time gives distance, not acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In lab notebook case 628513, check dimensional consistency for a = v * t. The left side has dimensions m/s^2 and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity times time gives distance, not acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


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

```python
def clamp_006892(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_006892"}

Answer: python_code

Prompt: Write a Python function clamp_006892(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_006892(1, 3, 2)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 17.25 kg and a = 5.9 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 101.775, "unit": "N"}

Answer: 101.775

Prompt: A cart has mass 17.25 kg and measured acceleration 5.9 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 17.25 kg and a = 5.9 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 = 33.2^2 and multiply by 0.5 * 4.35.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2397.372, "unit": "J"}

Answer: 2397.372

Prompt: A body of mass 4.35 kg moves at 33.2 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 33.2^2 and multiply by 0.5 * 4.35.
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 = 1230 * 3.495 * 33.
Final: {"answer": 141862.05, "unit": "J"}

Answer: 141862.05

Prompt: A 1230 g sample has specific heat 3.495 J/(g*K). How much heat is needed to raise its temperature by 33 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1230 * 3.495 * 33.


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=85.4 V and R=18.5 ohm.
Final: {"answer": 394.2249, "unit": "W"}

Answer: 394.2249

Prompt: A resistor has resistance 18.5 ohm and voltage 85.4 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=85.4 V and R=18.5 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=788 mg, t=105.5 h, and T_half=87.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 341.6411, "unit": "mg"}

Answer: 341.6411

Prompt: A radioactive sample starts with 788 mg. Its half-life is 87.5 hours. How many mg remain after 105.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=788 mg, t=105.5 h, and T_half=87.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.79 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 140.3368, "unit": "g"}

Answer: 140.3368

Prompt: A sample contains 7.79 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 7.79 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 945 mL to 0.945 L.
3. Compute M = 2.11 / 0.945.
Final: {"answer": 2.2328, "unit": "mol/L"}

Answer: 2.2328

Prompt: A solution contains 2.11 mol of solute in 945 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 945 mL to 0.945 L.
Compute M = 2.11 / 0.945.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-2.3)=19.98 and f(2.9)=1.26.
3. The interval width is 5.2.
Final: {"answer": 55.224, "unit": "area_units"}

Answer: 55.224

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.6x+11.7 from x=-2.3 to x=2.9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-2.3)=19.98 and f(2.9)=1.26.
The interval width is 5.2.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.06 and add it to the current value.
Final: {"answer": [-0.891, -1.094, 1.111], "unit": "temperature_units"}

Answer: -0.891

Answer: -1.094

Answer: 1.111

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=1.8, right=2.9; current interior values are [-1.05, -1.25, 1.15]. Use r=0.06. 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=0.06 and add it to the current value.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 24238.0 / 1000.
Final: {"answer": 24.238, "unit": "kg"}

Answer: 24.238

Prompt: Convert 24238 g to kg. Return the converted value with unit.

There are 1000 grams in one kilogram.
Apply the conversion factor directly to the given value.
Compute 24238.0 / 1000.


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": [-4.6, -2.75], "unit": "dimensionless"}

Answer: -4.6

Answer: -2.75

Prompt: A line passes through points (14, -67.15) and (16.5, -78.65). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. 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_006910(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_006910"}

Answer: python_code

Prompt: Write a Python function unit_vector_006910(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_006910(0, 0)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 26.85^2 and multiply by 0.5 * 19.3.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 6956.9021, "unit": "J"}

Answer: 6956.9021

Prompt: A body of mass 19.3 kg moves at 26.85 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 26.85^2 and multiply by 0.5 * 19.3.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=9 degrees through the sine of 2 theta.
3. Substitute v=53.7 m/s and g=9.81 m/s^2.
Final: {"answer": 90.8368, "unit": "m"}

Answer: 90.8368

Prompt: A projectile is launched on level ground with no air resistance at speed 53.7 m/s and angle 9 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=9 degrees through the sine of 2 theta.
Substitute v=53.7 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.62 * 8.314 * 559 / 92.5.
Final: {"answer": 232.1251, "unit": "kPa"}

Answer: 232.1251

Prompt: An ideal gas sample has n=4.62 mol, T=559 K, and V=92.5 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 = 4.62 * 8.314 * 559 / 92.5.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 525 * 3.35 * 92.
Final: {"answer": 161805.0, "unit": "J"}

Answer: 161805.0

Prompt: A 525 g sample has specific heat 3.35 J/(g*K). How much heat is needed to raise its temperature by 92 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 525 * 3.35 * 92.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=127.4 V and R=144.3 ohm.
Final: {"answer": 112.4793, "unit": "W"}

Answer: 112.4793

Prompt: A resistor has resistance 144.3 ohm and voltage 127.4 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=127.4 V and R=144.3 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=417.5 mg, t=42.5 h, and T_half=73.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 279.635, "unit": "mg"}

Answer: 279.635

Prompt: A radioactive sample starts with 417.5 mg. Its half-life is 73.5 hours. How many mg remain after 42.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=417.5 mg, t=42.5 h, and T_half=73.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 6.44 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 376.3536, "unit": "g"}

Answer: 376.3536

Prompt: A sample contains 6.44 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 = 6.44 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.5 y, so each step multiplies y by 1 - h*0.5.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 0.0, "unit": "state_units"}

Answer: 0.0

Prompt: Use explicit Euler on dy/dt = -0.5 y with y(0)=24.5, step size h=1.5, for 12 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.5 y, so each step multiplies y by 1 - h*0.5.
Apply that update for 12 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(3.1)=-27.96 and f(12.4)=-89.34.
3. The interval width is 9.3.
Final: {"answer": -545.445, "unit": "area_units"}

Answer: -545.445

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-6.6x+-7.5 from x=3.1 to x=12.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(3.1)=-27.96 and f(12.4)=-89.34.
The interval width is 9.3.


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": [2.326, 1.601, 0.863], "unit": "temperature_units"}

Answer: 2.326

Answer: 1.601

Answer: 0.863

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=1.9, right=-1.65; current interior values are [2.35, 1.6, 0.9]. 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. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 370.0 / 100.
Final: {"answer": 3.7, "unit": "m"}

Answer: 3.7

Prompt: Convert 370 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 370.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=316 degrees and keep the unit as newtons.
Final: {"answer": [254.2866, -245.5617], "unit": "N"}

Answer: 254.2866

Answer: -245.5617

Prompt: A vector has magnitude 353.5 N at 316 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=316 degrees and keep the unit as newtons.