Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1305 * 2.385 * 29.
Final: {"answer": 90260.325, "unit": "J"}

Answer: 90260.325

Prompt: A 1305 g sample has specific heat 2.385 J/(g*K). How much heat is needed to raise its temperature by 29 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1305 * 2.385 * 29.


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=204 V and R=165.1 ohm.
Final: {"answer": 252.0654, "unit": "W"}

Answer: 252.0654

Prompt: A resistor has resistance 165.1 ohm and voltage 204 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=204 V and R=165.1 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.02 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 2.0017, "unit": "g"}

Answer: 2.0017

Prompt: A sample contains 0.02 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.02 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4355 mL to 4.355 L.
3. Compute M = 1.775 / 4.355.
Final: {"answer": 0.4076, "unit": "mol/L"}

Answer: 0.4076

Prompt: A solution contains 1.775 mol of solute in 4355 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4355 mL to 4.355 L.
Compute M = 1.775 / 4.355.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.77 y, so each step multiplies y by 1 - h*0.77.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": 0.0111, "unit": "state_units"}

Answer: 0.0111

Prompt: Use explicit Euler on dy/dt = -0.77 y with y(0)=43, step size h=0.9, for 7 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.77 y, so each step multiplies y by 1 - h*0.77.
Apply that update for 7 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(7.4)=-16.18 and f(17.2)=-42.64.
3. The interval width is 9.8.
Final: {"answer": -288.218, "unit": "area_units"}

Answer: -288.218

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-2.7x+3.8 from x=7.4 to x=17.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(7.4)=-16.18 and f(17.2)=-42.64.
The interval width is 9.8.

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


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.16 and add it to the current value.
Final: {"answer": [2.176, 1.536, -0.756], "unit": "temperature_units"}

Answer: 2.176

Answer: 1.536

Answer: -0.756

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=-0.3, right=0.5; current interior values are [2.8, 2.0, -1.7]. Use r=0.16. 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=0.16 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 220.0 + 273.15.
Final: {"answer": 493.15, "unit": "K"}

Answer: 493.15

Prompt: Convert 220 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 220.0 + 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=135 degrees and keep the unit as newtons.
Final: {"answer": [-290.9744, 290.9744], "unit": "N"}

Answer: -290.9744

Answer: 290.9744

Prompt: A vector has magnitude 411.5 N at 135 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=135 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": [-7.55, -28.5], "unit": "dimensionless"}

Answer: -7.55

Answer: -28.5

Prompt: A line passes through points (17.75, -162.512) and (30.5, -258.775). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: inconsistent

Prompt: In sensor calibration case 294553, check dimensional consistency for C = n * V. The left side has dimensions mol/L and the right side has dimensions mol*L. Return consistent or inconsistent with a brief reason.

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

Answer: inconsistent


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

```python
import math

def unit_vector_001816(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_001816"}

Answer: python_code

Prompt: Write a Python function unit_vector_001816(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_001816(0, 0)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 9.5 kg and a = 0.75 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 7.125, "unit": "N"}

Answer: 7.125

Prompt: A cart has mass 9.5 kg and measured acceleration 0.75 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 9.5 kg and a = 0.75 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=57 degrees through the sine of 2 theta.
3. Substitute v=9.4 m/s and g=9.81 m/s^2.
Final: {"answer": 8.2284, "unit": "m"}

Answer: 8.2284

Prompt: A projectile is launched on level ground with no air resistance at speed 9.4 m/s and angle 57 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=57 degrees through the sine of 2 theta.
Substitute v=9.4 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 = 5.1 * 8.314 * 555 / 75.4.
Final: {"answer": 312.1058, "unit": "kPa"}

Answer: 312.1058

Prompt: An ideal gas sample has n=5.1 mol, T=555 K, and V=75.4 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 = 5.1 * 8.314 * 555 / 75.4.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1930 * 1.22 * 39.
Final: {"answer": 91829.4, "unit": "J"}

Answer: 91829.4

Prompt: A 1930 g sample has specific heat 1.22 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 = 1930 * 1.22 * 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=82.3 V and R=19.3 ohm.
Final: {"answer": 350.9477, "unit": "W"}

Answer: 350.9477

Prompt: A resistor has resistance 19.3 ohm and voltage 82.3 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=82.3 V and R=19.3 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=695.5 mg, t=254 h, and T_half=79.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 75.9479, "unit": "mg"}

Answer: 75.9479

Prompt: A radioactive sample starts with 695.5 mg. Its half-life is 79.5 hours. How many mg remain after 254 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=695.5 mg, t=254 h, and T_half=79.5 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 = 2.23 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 223.194, "unit": "g"}

Answer: 223.194

Prompt: A sample contains 2.23 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 2.23 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3010 mL to 3.01 L.
3. Compute M = 4.425 / 3.01.
Final: {"answer": 1.4701, "unit": "mol/L"}

Answer: 1.4701

Prompt: A solution contains 4.425 mol of solute in 3010 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3010 mL to 3.01 L.
Compute M = 4.425 / 3.01.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-7.7)=13.07 and f(-0.6)=5.26.
3. The interval width is 7.1.
Final: {"answer": 65.0715, "unit": "area_units"}

Answer: 65.0715

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-1.1x+4.6 from x=-7.7 to x=-0.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-7.7)=13.07 and f(-0.6)=5.26.
The interval width is 7.1.


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=0.2 and add it to the current value.
Final: {"answer": [0.78, -0.89, -0.82], "unit": "temperature_units"}

Answer: 0.78

Answer: -0.89

Answer: -0.82

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.75, right=0.2; current interior values are [1.25, -1.6, -0.9]. Use r=0.2. 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=0.2 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=57 degrees and keep the unit as newtons.
Final: {"answer": [59.0933, 90.9958], "unit": "N"}

Answer: 59.0933

Answer: 90.9958

Prompt: A vector has magnitude 108.5 N at 57 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=57 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": [-5.8, -26.0], "unit": "dimensionless"}

Answer: -5.8

Answer: -26.0

Prompt: A line passes through points (14.25, -108.65) and (24.25, -166.65). 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. 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 simulation spec case 652431, 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. 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_001834(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_001834"}

Answer: python_code

Prompt: Write a Python function normalize_001834(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_001834([2, 3, 5]); assert len(out) == 3

try:
    normalize_001834([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: 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)=46.5 and h=0.5, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 11.05 kg and a = 8.5 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 93.925, "unit": "N"}

Answer: 93.925

Prompt: A cart has mass 11.05 kg and measured acceleration 8.5 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 11.05 kg and a = 8.5 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=61 degrees through the sine of 2 theta.
3. Substitute v=47.2 m/s and g=9.81 m/s^2.
Final: {"answer": 192.5908, "unit": "m"}

Answer: 192.5908

Prompt: A projectile is launched on level ground with no air resistance at speed 47.2 m/s and angle 61 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=61 degrees through the sine of 2 theta.
Substitute v=47.2 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 = 4.83 * 8.314 * 433 / 62.8.
Final: {"answer": 276.8761, "unit": "kPa"}

Answer: 276.8761

Prompt: An ideal gas sample has n=4.83 mol, T=433 K, and V=62.8 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.83 * 8.314 * 433 / 62.8.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 75 * 4.41 * 98.
Final: {"answer": 32413.5, "unit": "J"}

Answer: 32413.5

Prompt: A 75 g sample has specific heat 4.41 J/(g*K). How much heat is needed to raise its temperature by 98 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 75 * 4.41 * 98.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=429 mg, t=156 h, and T_half=83.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 117.5044, "unit": "mg"}

Answer: 117.5044

Prompt: A radioactive sample starts with 429 mg. Its half-life is 83.5 hours. How many mg remain after 156 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=429 mg, t=156 h, and T_half=83.5 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 = 3.32 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 598.1179, "unit": "g"}

Answer: 598.1179

Prompt: A sample contains 3.32 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 3.32 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3645 mL to 3.645 L.
3. Compute M = 4.955 / 3.645.
Final: {"answer": 1.3594, "unit": "mol/L"}

Answer: 1.3594

Prompt: A solution contains 4.955 mol of solute in 3645 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3645 mL to 3.645 L.
Compute M = 4.955 / 3.645.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.53 y, so each step multiplies y by 1 - h*0.53.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 0.0134, "unit": "state_units"}

Answer: 0.0134

Prompt: Use explicit Euler on dy/dt = -0.53 y with y(0)=32, step size h=0.9, 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) = -0.53 y, so each step multiplies y by 1 - h*0.53.
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(-7.6)=26.76 and f(1.4)=-23.64.
3. The interval width is 9.
Final: {"answer": 14.04, "unit": "area_units"}

Answer: 14.04

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-5.6x+-15.8 from x=-7.6 to x=1.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-7.6)=26.76 and f(1.4)=-23.64.
The interval width is 9.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.14 and add it to the current value.
Final: {"answer": [1.508, 1.099, -0.732], "unit": "temperature_units"}

Answer: 1.508

Answer: 1.099

Answer: -0.732

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.4, right=-2; current interior values are [1.55, 1.4, -0.9]. Use r=0.14. Return the next interior vector.

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


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=194 degrees and keep the unit as newtons.
Final: {"answer": [-164.4651, -41.0058], "unit": "N"}

Answer: -164.4651

Answer: -41.0058

Prompt: A vector has magnitude 169.5 N at 194 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=194 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": [-6.35, 25.0], "unit": "dimensionless"}

Answer: -6.35

Answer: 25.0

Prompt: A line passes through points (-17.5, 136.125) and (-13.5, 110.725). 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. mass times acceleration has newton dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In student derivation case 464965, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
mass times acceleration has newton dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
import math

def unit_vector_001852(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_001852"}

Answer: python_code

Prompt: Write a Python function unit_vector_001852(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_001852(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: target_schema, valid_examples.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["target_schema", "valid_examples"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a parser that converts each mobile survey record from batch 721929 into the correct schema, but the prompt does not include the target schema or examples. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 14.5 kg and a = 13.45 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 195.025, "unit": "N"}

Answer: 195.025

Prompt: A cart has mass 14.5 kg and measured acceleration 13.45 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 = 14.5 kg and a = 13.45 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 5.5^2 and multiply by 0.5 * 23.8.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 359.975, "unit": "J"}

Answer: 359.975

Prompt: A body of mass 23.8 kg moves at 5.5 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 5.5^2 and multiply by 0.5 * 23.8.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=23 degrees through the sine of 2 theta.
3. Substitute v=77.4 m/s and g=9.81 m/s^2.
Final: {"answer": 439.2856, "unit": "m"}

Answer: 439.2856

Prompt: A projectile is launched on level ground with no air resistance at speed 77.4 m/s and angle 23 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=23 degrees through the sine of 2 theta.
Substitute v=77.4 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.07 * 8.314 * 444 / 117.6.
Final: {"answer": 159.1452, "unit": "kPa"}

Answer: 159.1452

Prompt: An ideal gas sample has n=5.07 mol, T=444 K, and V=117.6 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 = 5.07 * 8.314 * 444 / 117.6.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 465 * 4.31 * 112.
Final: {"answer": 224464.8, "unit": "J"}

Answer: 224464.8

Prompt: A 465 g sample has specific heat 4.31 J/(g*K). How much heat is needed to raise its temperature by 112 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 465 * 4.31 * 112.


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=37.4 V and R=158.8 ohm.
Final: {"answer": 8.8083, "unit": "W"}

Answer: 8.8083

Prompt: A resistor has resistance 158.8 ohm and voltage 37.4 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=37.4 V and R=158.8 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=711.5 mg, t=349 h, and T_half=43.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 2.7354, "unit": "mg"}

Answer: 2.7354

Prompt: A radioactive sample starts with 711.5 mg. Its half-life is 43.5 hours. How many mg remain after 349 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=711.5 mg, t=349 h, and T_half=43.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2140 mL to 2.14 L.
3. Compute M = 1.3 / 2.14.
Final: {"answer": 0.6075, "unit": "mol/L"}

Answer: 0.6075

Prompt: A solution contains 1.3 mol of solute in 2140 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2140 mL to 2.14 L.
Compute M = 1.3 / 2.14.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.16 y, so each step multiplies y by 1 - h*0.16.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 0.4951, "unit": "state_units"}

Answer: 0.4951

Prompt: Use explicit Euler on dy/dt = -0.16 y with y(0)=4.5, step size h=1.05, 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) = -0.16 y, so each step multiplies y by 1 - h*0.16.
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(-9.5)=66.9 and f(-9.2)=64.5.
3. The interval width is 0.3.
Final: {"answer": 19.71, "unit": "area_units"}

Answer: 19.71

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-8x+-9.1 from x=-9.5 to x=-9.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-9.5)=66.9 and f(-9.2)=64.5.
The interval width is 0.3.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 44105.0 / 1000.
Final: {"answer": 44.105, "unit": "kg"}

Answer: 44.105

Prompt: Convert 44105 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 44105.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=144 degrees and keep the unit as newtons.
Final: {"answer": [-95.464, 69.3587], "unit": "N"}

Answer: -95.464

Answer: 69.3587

Prompt: A vector has magnitude 118 N at 144 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=144 degrees and keep the unit as newtons.


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": [-2.2, 25.25], "unit": "dimensionless"}

Answer: -2.2

Answer: 25.25

Prompt: A line passes through points (10.5, 2.15) and (23.5, -26.45). 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 instrument manual case 62965, 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 running_total_001870(values):
    total = 0
    out = []
    for value in values:
        total += value
        out.append(total)
    return out
```
Final: {"answer": "python_code", "function": "running_total_001870"}

Answer: python_code

Prompt: Write a Python function running_total_001870(values) that returns cumulative sums as a new list. It must not mutate the input list. Return a single fenced Python code block.

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


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

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a segmentation scoring problem with 2800 items and budget 55185, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 12.45^2 and multiply by 0.5 * 8.4.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 651.0105, "unit": "J"}

Answer: 651.0105

Prompt: A body of mass 8.4 kg moves at 12.45 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 12.45^2 and multiply by 0.5 * 8.4.
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=76 degrees through the sine of 2 theta.
3. Substitute v=36.9 m/s and g=9.81 m/s^2.
Final: {"answer": 65.1618, "unit": "m"}

Answer: 65.1618

Prompt: A projectile is launched on level ground with no air resistance at speed 36.9 m/s and angle 76 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=76 degrees through the sine of 2 theta.
Substitute v=36.9 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 = 4.22 * 8.314 * 447 / 65.8.
Final: {"answer": 238.3439, "unit": "kPa"}

Answer: 238.3439

Prompt: An ideal gas sample has n=4.22 mol, T=447 K, and V=65.8 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.22 * 8.314 * 447 / 65.8.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1700 * 1.805 * 110.
Final: {"answer": 337535.0, "unit": "J"}

Answer: 337535.0

Prompt: A 1700 g sample has specific heat 1.805 J/(g*K). How much heat is needed to raise its temperature by 110 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1700 * 1.805 * 110.


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=41.8 V and R=91.5 ohm.
Final: {"answer": 19.0955, "unit": "W"}

Answer: 19.0955

Prompt: A resistor has resistance 91.5 ohm and voltage 41.8 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=41.8 V and R=91.5 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=488 mg, t=32 h, and T_half=8.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 35.903, "unit": "mg"}

Answer: 35.903

Prompt: A radioactive sample starts with 488 mg. Its half-life is 8.5 hours. How many mg remain after 32 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=488 mg, t=32 h, and T_half=8.5 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 = 3.04 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 304.2645, "unit": "g"}

Answer: 304.2645

Prompt: A sample contains 3.04 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 3.04 mol * 100.087 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) = -0.53 y, so each step multiplies y by 1 - h*0.53.
3. Apply that update for 6 steps and round the final state.
Final: {"answer": 30.2935, "unit": "state_units"}

Answer: 30.2935

Prompt: Use explicit Euler on dy/dt = -0.53 y with y(0)=42, step size h=0.1, for 6 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.53 y, so each step multiplies y by 1 - h*0.53.
Apply that update for 6 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(-6.2)=59.1 and f(-0.5)=16.35.
3. The interval width is 5.7.
Final: {"answer": 215.0325, "unit": "area_units"}

Answer: 215.0325

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-7.5x+12.6 from x=-6.2 to x=-0.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-6.2)=59.1 and f(-0.5)=16.35.
The interval width is 5.7.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=65 degrees and keep the unit as newtons.
Final: {"answer": [39.0922, 83.8335], "unit": "N"}

Answer: 39.0922

Answer: 83.8335

Prompt: A vector has magnitude 92.5 N at 65 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=65 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.1, 20.0], "unit": "dimensionless"}

Answer: 8.1

Answer: 20.0

Prompt: A line passes through points (16.75, 155.675) and (23.75, 212.375). 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. mass times acceleration has newton dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In quality-control check case 220389, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
mass times acceleration has newton dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
def pairwise_ratios_001888(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_001888"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_001888(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_001888([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_001888([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: volume.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["volume"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute pressure from the ideal gas law. The prompt gives n=0.61 mol and T=613 K, but does not give volume. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 5.85 kg and a = 6.9 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 40.365, "unit": "N"}

Answer: 40.365

Prompt: A cart has mass 5.85 kg and measured acceleration 6.9 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 = 5.85 kg and a = 6.9 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=26 degrees through the sine of 2 theta.
3. Substitute v=81.8 m/s and g=9.81 m/s^2.
Final: {"answer": 537.4892, "unit": "m"}

Answer: 537.4892

Prompt: A projectile is launched on level ground with no air resistance at speed 81.8 m/s and angle 26 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=26 degrees through the sine of 2 theta.
Substitute v=81.8 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 1.25 * 8.314 * 495 / 102.9.
Final: {"answer": 49.9931, "unit": "kPa"}

Answer: 49.9931

Prompt: An ideal gas sample has n=1.25 mol, T=495 K, and V=102.9 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 = 1.25 * 8.314 * 495 / 102.9.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 650 * 2.08 * 45.
Final: {"answer": 60840.0, "unit": "J"}

Answer: 60840.0

Prompt: A 650 g sample has specific heat 2.08 J/(g*K). How much heat is needed to raise its temperature by 45 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 650 * 2.08 * 45.


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=231.9 V and R=113.4 ohm.
Final: {"answer": 474.2294, "unit": "W"}

Answer: 474.2294

Prompt: A resistor has resistance 113.4 ohm and voltage 231.9 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=231.9 V and R=113.4 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=33.5 mg, t=302 h, and T_half=48 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.4276, "unit": "mg"}

Answer: 0.4276

Prompt: A radioactive sample starts with 33.5 mg. Its half-life is 48 hours. How many mg remain after 302 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=33.5 mg, t=302 h, and T_half=48 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 = 7.04 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 126.8256, "unit": "g"}

Answer: 126.8256

Prompt: A sample contains 7.04 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 = 7.04 mol * 18.015 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) = -0.39 y, so each step multiplies y by 1 - h*0.39.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": 6.8269, "unit": "state_units"}

Answer: 6.8269

Prompt: Use explicit Euler on dy/dt = -0.39 y with y(0)=37, step size h=0.55, for 7 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.39 y, so each step multiplies y by 1 - h*0.39.
Apply that update for 7 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-1.8)=13.9 and f(2.4)=-2.9.
3. The interval width is 4.2.
Final: {"answer": 23.1, "unit": "area_units"}

Answer: 23.1

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-4x+6.7 from x=-1.8 to x=2.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-1.8)=13.9 and f(2.4)=-2.9.
The interval width is 4.2.


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=0.38 and add it to the current value.
Final: {"answer": [1.209, 0.627, 1.57], "unit": "temperature_units"}

Answer: 1.209

Answer: 0.627

Answer: 1.57

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2.2, right=2.55; current interior values are [0.05, 0.95, 1.0]. 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=19 degrees and keep the unit as newtons.
Final: {"answer": [72.3322, 24.906], "unit": "N"}

Answer: 72.3322

Answer: 24.906

Prompt: A vector has magnitude 76.5 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. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.1, 26.5], "unit": "dimensionless"}

Answer: 8.1

Answer: 26.5

Prompt: A line passes through points (9.25, 101.425) and (11.5, 119.65). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: consistent

Prompt: In quality-control check case 542014, 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. 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=0.26 and interior values [-1.4, 2.75, -1.2], 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 = 11.05 kg and a = 11.15 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 123.2075, "unit": "N"}

Answer: 123.2075

Prompt: A cart has mass 11.05 kg and measured acceleration 11.15 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 = 11.05 kg and a = 11.15 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 = 12.25^2 and multiply by 0.5 * 14.4.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1080.45, "unit": "J"}

Answer: 1080.45

Prompt: A body of mass 14.4 kg moves at 12.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 12.25^2 and multiply by 0.5 * 14.4.
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=59 degrees through the sine of 2 theta.
3. Substitute v=33.8 m/s and g=9.81 m/s^2.
Final: {"answer": 102.8251, "unit": "m"}

Answer: 102.8251

Prompt: A projectile is launched on level ground with no air resistance at speed 33.8 m/s and angle 59 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=59 degrees through the sine of 2 theta.
Substitute v=33.8 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.87 * 8.314 * 568 / 7.6.
Final: {"answer": 3647.3956, "unit": "kPa"}

Answer: 3647.3956

Prompt: An ideal gas sample has n=5.87 mol, T=568 K, and V=7.6 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 = 5.87 * 8.314 * 568 / 7.6.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=193.7 V and R=59.1 ohm.
Final: {"answer": 634.8509, "unit": "W"}

Answer: 634.8509

Prompt: A resistor has resistance 59.1 ohm and voltage 193.7 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=193.7 V and R=59.1 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=570.5 mg, t=225.5 h, and T_half=49.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 24.2603, "unit": "mg"}

Answer: 24.2603

Prompt: A radioactive sample starts with 570.5 mg. Its half-life is 49.5 hours. How many mg remain after 225.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=570.5 mg, t=225.5 h, and T_half=49.5 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 = 6.19 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 361.7436, "unit": "g"}

Answer: 361.7436

Prompt: A sample contains 6.19 mol of sodium chloride. Using molar mass 58.44 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.19 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 1925 mL to 1.925 L.
3. Compute M = 4.315 / 1.925.
Final: {"answer": 2.2416, "unit": "mol/L"}

Answer: 2.2416

Prompt: A solution contains 4.315 mol of solute in 1925 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1925 mL to 1.925 L.
Compute M = 4.315 / 1.925.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.02 y, so each step multiplies y by 1 - h*0.02.
3. Apply that update for 6 steps and round the final state.
Final: {"answer": 5.6832, "unit": "state_units"}

Answer: 5.6832

Prompt: Use explicit Euler on dy/dt = -0.02 y with y(0)=6, step size h=0.45, for 6 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.02 y, so each step multiplies y by 1 - h*0.02.
Apply that update for 6 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(7)=42.7 and f(8.4)=49.7.
3. The interval width is 1.4.
Final: {"answer": 64.68, "unit": "area_units"}

Answer: 64.68

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=5x+7.7 from x=7 to x=8.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(7)=42.7 and f(8.4)=49.7.
The interval width is 1.4.


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=0.34 and add it to the current value.
Final: {"answer": [0.305, 0.72, 0.363], "unit": "temperature_units"}

Answer: 0.305

Answer: 0.72

Answer: 0.363

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.25, right=0.95; current interior values are [1.75, -2.0, 2.25]. Use r=0.34. 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=0.34 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 250.0 / 100.
Final: {"answer": 2.5, "unit": "m"}

Answer: 2.5

Prompt: Convert 250 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 250.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=60 degrees and keep the unit as newtons.
Final: {"answer": [190.0, 329.0897], "unit": "N"}

Answer: 190.0

Answer: 329.0897

Prompt: A vector has magnitude 380 N at 60 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=60 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-1.15, 14.5], "unit": "dimensionless"}

Answer: -1.15

Answer: 14.5

Prompt: A line passes through points (-10.25, 26.2875) and (0.5, 13.925). Return [slope, intercept] for y = slope*x + intercept.

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


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 lab notebook case 274577, 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_001924(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_001924"}

Answer: python_code

Prompt: Write a Python function safe_mean_001924(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_001924([])
    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: volume.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["volume"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute pressure from the ideal gas law. The prompt gives n=5.61 mol and T=612 K, but does not give volume. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 20.7 kg and a = 3.65 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 75.555, "unit": "N"}

Answer: 75.555

Prompt: A cart has mass 20.7 kg and measured acceleration 3.65 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 20.7 kg and a = 3.65 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 = 11.15^2 and multiply by 0.5 * 20.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1258.7653, "unit": "J"}

Answer: 1258.7653

Prompt: A body of mass 20.25 kg moves at 11.15 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 11.15^2 and multiply by 0.5 * 20.25.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=64 degrees through the sine of 2 theta.
3. Substitute v=49.4 m/s and g=9.81 m/s^2.
Final: {"answer": 196.0275, "unit": "m"}

Answer: 196.0275

Prompt: A projectile is launched on level ground with no air resistance at speed 49.4 m/s and angle 64 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=64 degrees through the sine of 2 theta.
Substitute v=49.4 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.11 * 8.314 * 446 / 56.0.
Final: {"answer": 272.1439, "unit": "kPa"}

Answer: 272.1439

Prompt: An ideal gas sample has n=4.11 mol, T=446 K, and V=56.0 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.11 * 8.314 * 446 / 56.0.


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=81.6 V and R=22 ohm.
Final: {"answer": 302.6618, "unit": "W"}

Answer: 302.6618

Prompt: A resistor has resistance 22 ohm and voltage 81.6 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=81.6 V and R=22 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=93.5 mg, t=178 h, and T_half=90 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 23.7378, "unit": "mg"}

Answer: 23.7378

Prompt: A radioactive sample starts with 93.5 mg. Its half-life is 90 hours. How many mg remain after 178 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=93.5 mg, t=178 h, and T_half=90 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 = 1.97 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 86.6997, "unit": "g"}

Answer: 86.6997

Prompt: A sample contains 1.97 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 1.97 mol * 44.01 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.44 and add it to the current value.
Final: {"answer": [0.524, -0.956, 0.022], "unit": "temperature_units"}

Answer: 0.524

Answer: -0.956

Answer: 0.022

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=0.8; current interior values are [-0.95, -0.45, -1.1]. 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. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=82 degrees and keep the unit as newtons.
Final: {"answer": [15.3786, 109.4246], "unit": "N"}

Answer: 15.3786

Answer: 109.4246

Prompt: A vector has magnitude 110.5 N at 82 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=82 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": [-7.65, -9.5], "unit": "dimensionless"}

Answer: -7.65

Answer: -9.5

Prompt: A line passes through points (-19, 135.85) and (-9.5, 63.175). 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. mass times acceleration has newton dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In lab notebook case 693201, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
mass times acceleration has newton dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
def threshold_flags_001942(values, threshold):
    return [value >= threshold for value in values]
```
Final: {"answer": "python_code", "function": "threshold_flags_001942"}

Answer: python_code

Prompt: Write a Python function threshold_flags_001942(values, threshold) that returns a list of booleans showing whether each value is greater than or equal to threshold. Return a single fenced Python code block.

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

assert threshold_flags_001942([1, 2, 3], 2) == [False, True, True]

assert threshold_flags_001942([], 5) == []

assert threshold_flags_001942([-1, 0], 0) == [False, True]


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

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=27.5 and h=0.1, 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 = 15.05 kg and a = 7.55 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 113.6275, "unit": "N"}

Answer: 113.6275

Prompt: A cart has mass 15.05 kg and measured acceleration 7.55 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 15.05 kg and a = 7.55 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 20.6^2 and multiply by 0.5 * 14.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2970.52, "unit": "J"}

Answer: 2970.52

Prompt: A body of mass 14 kg moves at 20.6 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 20.6^2 and multiply by 0.5 * 14.
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=35 degrees through the sine of 2 theta.
3. Substitute v=59 m/s and g=9.81 m/s^2.
Final: {"answer": 333.4424, "unit": "m"}

Answer: 333.4424

Prompt: A projectile is launched on level ground with no air resistance at speed 59 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=59 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=18 mg, t=84 h, and T_half=73 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 8.1074, "unit": "mg"}

Answer: 8.1074

Prompt: A radioactive sample starts with 18 mg. Its half-life is 73 hours. How many mg remain after 84 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=18 mg, t=84 h, and T_half=73 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 = 7.96 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 350.3196, "unit": "g"}

Answer: 350.3196

Prompt: A sample contains 7.96 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 = 7.96 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 4830 mL to 4.83 L.
3. Compute M = 2.25 / 4.83.
Final: {"answer": 0.4658, "unit": "mol/L"}

Answer: 0.4658

Prompt: A solution contains 2.25 mol of solute in 4830 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4830 mL to 4.83 L.
Compute M = 2.25 / 4.83.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.42 y, so each step multiplies y by 1 - h*0.42.
3. Apply that update for 5 steps and round the final state.
Final: {"answer": 2.8655, "unit": "state_units"}

Answer: 2.8655

Prompt: Use explicit Euler on dy/dt = -0.42 y with y(0)=19, step size h=0.75, 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) = -0.42 y, so each step multiplies y by 1 - h*0.42.
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(-3.2)=-8.14 and f(3.6)=-16.98.
3. The interval width is 6.8.
Final: {"answer": -85.408, "unit": "area_units"}

Answer: -85.408

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-1.3x+-12.3 from x=-3.2 to x=3.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-3.2)=-8.14 and f(3.6)=-16.98.
The interval width is 6.8.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.24 and add it to the current value.
Final: {"answer": [1.356, 0.888, 1.506], "unit": "temperature_units"}

Answer: 1.356

Answer: 0.888

Answer: 1.506

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=0.75, right=2.35; current interior values are [2.4, -0.3, 1.95]. Use r=0.24. 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=0.24 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=306 degrees and keep the unit as newtons.
Final: {"answer": [56.1335, -77.2611], "unit": "N"}

Answer: 56.1335

Answer: -77.2611

Prompt: A vector has magnitude 95.5 N at 306 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=306 degrees and keep the unit as newtons.


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

```python
def weighted_sum_001960(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_001960"}

Answer: python_code

Prompt: Write a Python function weighted_sum_001960(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_001960([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)=25.5 and h=1.85, 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 = 2.05 kg and a = 6 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 12.3, "unit": "N"}

Answer: 12.3

Prompt: A cart has mass 2.05 kg and measured acceleration 6 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 2.05 kg and a = 6 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 = 7.85^2 and multiply by 0.5 * 19.6.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 603.9005, "unit": "J"}

Answer: 603.9005

Prompt: A body of mass 19.6 kg moves at 7.85 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 7.85^2 and multiply by 0.5 * 19.6.
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=75 degrees through the sine of 2 theta.
3. Substitute v=74.3 m/s and g=9.81 m/s^2.
Final: {"answer": 281.3705, "unit": "m"}

Answer: 281.3705

Prompt: A projectile is launched on level ground with no air resistance at speed 74.3 m/s and angle 75 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=75 degrees through the sine of 2 theta.
Substitute v=74.3 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 = 4.07 * 8.314 * 268 / 32.4.
Final: {"answer": 279.8944, "unit": "kPa"}

Answer: 279.8944

Prompt: An ideal gas sample has n=4.07 mol, T=268 K, and V=32.4 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 = 4.07 * 8.314 * 268 / 32.4.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 640 * 2.43 * 118.
Final: {"answer": 183513.6, "unit": "J"}

Answer: 183513.6

Prompt: A 640 g sample has specific heat 2.43 J/(g*K). How much heat is needed to raise its temperature by 118 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 640 * 2.43 * 118.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=615.5 mg, t=144.5 h, and T_half=15 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.775, "unit": "mg"}

Answer: 0.775

Prompt: A radioactive sample starts with 615.5 mg. Its half-life is 15 hours. How many mg remain after 144.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=615.5 mg, t=144.5 h, and T_half=15 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 4485 mL to 4.485 L.
3. Compute M = 4.065 / 4.485.
Final: {"answer": 0.9064, "unit": "mol/L"}

Answer: 0.9064

Prompt: A solution contains 4.065 mol of solute in 4485 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4485 mL to 4.485 L.
Compute M = 4.065 / 4.485.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.42 y, so each step multiplies y by 1 - h*0.42.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": 0.5665, "unit": "state_units"}

Answer: 0.5665

Prompt: Use explicit Euler on dy/dt = -0.42 y with y(0)=39, step size h=1.8, 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) = -0.42 y, so each step multiplies y by 1 - h*0.42.
Apply that update for 3 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=0.46 and add it to the current value.
Final: {"answer": [0.398, -0.642, -0.702], "unit": "temperature_units"}

Answer: 0.398

Answer: -0.642

Answer: -0.702

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=0.3, right=-1.9; current interior values are [-0.2, 0.6, -1.3]. Use r=0.46. 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=0.46 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 3836.5 / 100.
Final: {"answer": 38.365, "unit": "m"}

Answer: 38.365

Prompt: Convert 3836.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 3836.5 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=129 degrees and keep the unit as newtons.
Final: {"answer": [-226.5553, 279.7725], "unit": "N"}

Answer: -226.5553

Answer: 279.7725

Prompt: A vector has magnitude 360 N at 129 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=129 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.2, -8.5], "unit": "dimensionless"}

Answer: 8.2

Answer: -8.5

Prompt: A line passes through points (-11.75, -104.85) and (1, -0.3). 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 sensor calibration case 560772, 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_001978(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_001978"}

Answer: python_code

Prompt: Write a Python function unit_vector_001978(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_001978(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: 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)=29 and h=1.65, 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 = 10.25 kg and a = 6.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 64.575, "unit": "N"}

Answer: 64.575

Prompt: A cart has mass 10.25 kg and measured acceleration 6.3 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 = 10.25 kg and a = 6.3 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 = 17^2 and multiply by 0.5 * 26.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 3793.125, "unit": "J"}

Answer: 3793.125

Prompt: A body of mass 26.25 kg moves at 17 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 17^2 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=79 degrees through the sine of 2 theta.
3. Substitute v=30.6 m/s and g=9.81 m/s^2.
Final: {"answer": 35.756, "unit": "m"}

Answer: 35.756

Prompt: A projectile is launched on level ground with no air resistance at speed 30.6 m/s and angle 79 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=79 degrees through the sine of 2 theta.
Substitute v=30.6 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 = 1.97 * 8.314 * 574 / 90.7.
Final: {"answer": 103.6528, "unit": "kPa"}

Answer: 103.6528

Prompt: An ideal gas sample has n=1.97 mol, T=574 K, and V=90.7 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 1.97 * 8.314 * 574 / 90.7.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1160 * 2.655 * 63.
Final: {"answer": 194027.4, "unit": "J"}

Answer: 194027.4

Prompt: A 1160 g sample has specific heat 2.655 J/(g*K). How much heat is needed to raise its temperature by 63 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1160 * 2.655 * 63.


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=36 V and R=36.6 ohm.
Final: {"answer": 35.4098, "unit": "W"}

Answer: 35.4098

Prompt: A resistor has resistance 36.6 ohm and voltage 36 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=36 V and R=36.6 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=459 mg, t=326.5 h, and T_half=82 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 29.0536, "unit": "mg"}

Answer: 29.0536

Prompt: A radioactive sample starts with 459 mg. Its half-life is 82 hours. How many mg remain after 326.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=459 mg, t=326.5 h, and T_half=82 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 3950 mL to 3.95 L.
3. Compute M = 0.04 / 3.95.
Final: {"answer": 0.0101, "unit": "mol/L"}

Answer: 0.0101

Prompt: A solution contains 0.04 mol of solute in 3950 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3950 mL to 3.95 L.
Compute M = 0.04 / 3.95.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.69 y, so each step multiplies y by 1 - h*0.69.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": -0.0001, "unit": "state_units"}

Answer: -0.0001

Prompt: Use explicit Euler on dy/dt = -0.69 y with y(0)=78, step size h=1.75, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.69 y, so each step multiplies y by 1 - h*0.69.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-0.3)=-9.29 and f(4)=-42.4.
3. The interval width is 4.3.
Final: {"answer": -111.1335, "unit": "area_units"}

Answer: -111.1335

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-7.7x+-11.6 from x=-0.3 to x=4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-0.3)=-9.29 and f(4)=-42.4.
The interval width is 4.3.


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=0.06 and add it to the current value.
Final: {"answer": [1.774, 1.127, 2.581], "unit": "temperature_units"}

Answer: 1.774

Answer: 1.127

Answer: 2.581

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=0.75, right=-1.2; current interior values are [1.9, 0.95, 2.95]. 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 12279.0 / 1000.
Final: {"answer": 12.279, "unit": "kg"}

Answer: 12.279

Prompt: Convert 12279 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 12279.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=180 degrees and keep the unit as newtons.
Final: {"answer": [-196.0, 0.0], "unit": "N"}

Answer: -196.0

Answer: 0.0

Prompt: A vector has magnitude 196 N at 180 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=180 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-8.6, -21.25], "unit": "dimensionless"}

Answer: -8.6

Answer: -21.25

Prompt: A line passes through points (-3.25, 6.7) and (10.25, -109.4). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: consistent

Prompt: In field worksheet case 326911, check dimensional consistency for Q = m * c * DeltaT. The left side has dimensions J and the right side has dimensions J. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a scheduler scoring problem with 3914 items and budget 19290, 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 = 5.9 kg and a = 15.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 88.795, "unit": "N"}

Answer: 88.795

Prompt: A cart has mass 5.9 kg and measured acceleration 15.05 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 = 5.9 kg and a = 15.05 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=8 degrees through the sine of 2 theta.
3. Substitute v=37.4 m/s and g=9.81 m/s^2.
Final: {"answer": 39.3018, "unit": "m"}

Answer: 39.3018

Prompt: A projectile is launched on level ground with no air resistance at speed 37.4 m/s and angle 8 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=8 degrees through the sine of 2 theta.
Substitute v=37.4 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 = 1000 * 4.185 * 48.
Final: {"answer": 200880.0, "unit": "J"}

Answer: 200880.0

Prompt: A 1000 g sample has specific heat 4.185 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 = 1000 * 4.185 * 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=211.7 V and R=133.4 ohm.
Final: {"answer": 335.9587, "unit": "W"}

Answer: 335.9587

Prompt: A resistor has resistance 133.4 ohm and voltage 211.7 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=211.7 V and R=133.4 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=159 mg, t=55.5 h, and T_half=37 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 56.215, "unit": "mg"}

Answer: 56.215

Prompt: A radioactive sample starts with 159 mg. Its half-life is 37 hours. How many mg remain after 55.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=159 mg, t=55.5 h, and T_half=37 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 = 1.71 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 171.1488, "unit": "g"}

Answer: 171.1488

Prompt: A sample contains 1.71 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 1.71 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3815 mL to 3.815 L.
3. Compute M = 4.175 / 3.815.
Final: {"answer": 1.0944, "unit": "mol/L"}

Answer: 1.0944

Prompt: A solution contains 4.175 mol of solute in 3815 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3815 mL to 3.815 L.
Compute M = 4.175 / 3.815.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.81 y, so each step multiplies y by 1 - h*0.81.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 2.5728, "unit": "state_units"}

Answer: 2.5728

Prompt: Use explicit Euler on dy/dt = -0.81 y with y(0)=31, step size h=0.25, 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) = -0.81 y, so each step multiplies y by 1 - h*0.81.
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(2.8)=-29.42 and f(7.8)=-53.92.
3. The interval width is 5.
Final: {"answer": -208.35, "unit": "area_units"}

Answer: -208.35

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-4.9x+-15.7 from x=2.8 to x=7.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(2.8)=-29.42 and f(7.8)=-53.92.
The interval width is 5.


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=0.48 and add it to the current value.
Final: {"answer": [2.054, 0.754, 1.53], "unit": "temperature_units"}

Answer: 2.054

Answer: 0.754

Answer: 1.53

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2.35, right=0.9; current interior values are [-1.45, 2.05, 2.85]. Use r=0.48. 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=0.48 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 -13.0 + 273.15.
Final: {"answer": 260.15, "unit": "K"}

Answer: 260.15

Prompt: Convert -13 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 -13.0 + 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=202 degrees and keep the unit as newtons.
Final: {"answer": [-426.041, -172.1317], "unit": "N"}

Answer: -426.041

Answer: -172.1317

Prompt: A vector has magnitude 459.5 N at 202 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=202 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": [-0.1, 0.25], "unit": "dimensionless"}

Answer: -0.1

Answer: 0.25

Prompt: A line passes through points (-12.5, 1.5) and (-7, 0.95). 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. mass times acceleration has newton dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In instrument manual case 667662, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
mass times acceleration has newton dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
def normalize_002014(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_002014"}

Answer: python_code

Prompt: Write a Python function normalize_002014(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_002014([2, 3, 5]); assert len(out) == 3

try:
    normalize_002014([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: launch_angle_or_vertical_velocity.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["launch_angle_or_vertical_velocity"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute projectile range on level ground. The prompt gives initial speed 53.2 m/s but does not give launch angle or vertical velocity. If there is insufficient information, say so explicitly.

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