Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(1.1)=-7.57 and f(6.4)=-27.18.
3. The interval width is 5.3.
Final: {"answer": -92.0875, "unit": "area_units"}

Answer: -92.0875

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3.7x+-3.5 from x=1.1 to x=6.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(1.1)=-7.57 and f(6.4)=-27.18.
The interval width is 5.3.


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=0.2 and add it to the current value.
Final: {"answer": [0.77, -0.62, 0.46], "unit": "temperature_units"}

Answer: 0.77

Answer: -0.62

Answer: 0.46

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=1.35, right=0.1; current interior values are [1.5, -2.0, 1.4]. 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. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 200.0 + 273.15.
Final: {"answer": 473.15, "unit": "K"}

Answer: 473.15

Prompt: Convert 200 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 200.0 + 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=211 degrees and keep the unit as newtons.
Final: {"answer": [-140.1469, -84.2087], "unit": "N"}

Answer: -140.1469

Answer: -84.2087

Prompt: A vector has magnitude 163.5 N at 211 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=211 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": [5.1, 26.75], "unit": "dimensionless"}

Answer: 5.1

Answer: 26.75

Prompt: A line passes through points (-13, -39.55) and (0, 26.75). 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 instrument manual case 678744, 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_002464(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_002464"}

Answer: python_code

Prompt: Write a Python function normalize_002464(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_002464([2, 3, 5]); assert len(out) == 3

try:
    normalize_002464([1, -1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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

Answer: INSUFFICIENT_INFO

Prompt: Solve the 1D heat equation after one finite-difference step. The prompt gives r=0.38 and interior values [1.0, -1.0, -0.85], 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 = 5.9 kg and a = 6.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 37.17, "unit": "N"}

Answer: 37.17

Prompt: A cart has mass 5.9 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 = 5.9 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 = 22.9^2 and multiply by 0.5 * 20.6.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 5401.423, "unit": "J"}

Answer: 5401.423

Prompt: A body of mass 20.6 kg moves at 22.9 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 22.9^2 and multiply by 0.5 * 20.6.
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=8 degrees through the sine of 2 theta.
3. Substitute v=8.2 m/s and g=9.81 m/s^2.
Final: {"answer": 1.8893, "unit": "m"}

Answer: 1.8893

Prompt: A projectile is launched on level ground with no air resistance at speed 8.2 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=8.2 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 = 2.96 * 8.314 * 296 / 30.2.
Final: {"answer": 241.2051, "unit": "kPa"}

Answer: 241.2051

Prompt: An ideal gas sample has n=2.96 mol, T=296 K, and V=30.2 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2.96 * 8.314 * 296 / 30.2.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=214.6 V and R=96.8 ohm.
Final: {"answer": 475.7558, "unit": "W"}

Answer: 475.7558

Prompt: A resistor has resistance 96.8 ohm and voltage 214.6 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=214.6 V and R=96.8 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=528.5 mg, t=42 h, and T_half=36 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 235.42, "unit": "mg"}

Answer: 235.42

Prompt: A radioactive sample starts with 528.5 mg. Its half-life is 36 hours. How many mg remain after 42 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=528.5 mg, t=42 h, and T_half=36 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 = 1.14 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 114.0992, "unit": "g"}

Answer: 114.0992

Prompt: A sample contains 1.14 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 = 1.14 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 3875 mL to 3.875 L.
3. Compute M = 1.995 / 3.875.
Final: {"answer": 0.5148, "unit": "mol/L"}

Answer: 0.5148

Prompt: A solution contains 1.995 mol of solute in 3875 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3875 mL to 3.875 L.
Compute M = 1.995 / 3.875.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.87 y, so each step multiplies y by 1 - h*0.87.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 0.4308, "unit": "state_units"}

Answer: 0.4308

Prompt: Use explicit Euler on dy/dt = -0.87 y with y(0)=12, step size h=0.3, 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) = -0.87 y, so each step multiplies y by 1 - h*0.87.
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(-4.6)=-2.94 and f(5.3)=-23.73.
3. The interval width is 9.9.
Final: {"answer": -132.0165, "unit": "area_units"}

Answer: -132.0165

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-2.1x+-12.6 from x=-4.6 to x=5.3. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.6)=-2.94 and f(5.3)=-23.73.
The interval width is 9.9.

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


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 176.0 + 273.15.
Final: {"answer": 449.15, "unit": "K"}

Answer: 449.15

Prompt: Convert 176 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 176.0 + 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=121 degrees and keep the unit as newtons.
Final: {"answer": [-250.8235, 417.4405], "unit": "N"}

Answer: -250.8235

Answer: 417.4405

Prompt: A vector has magnitude 487 N at 121 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=121 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": [-0.15, -15.0], "unit": "dimensionless"}

Answer: -0.15

Answer: -15.0

Prompt: A line passes through points (7, -16.05) and (10, -16.5). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. 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 lab notebook case 717643, 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_002482(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_002482"}

Answer: python_code

Prompt: Write a Python function unit_vector_002482(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_002482(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: 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=0.18 and interior values [-0.6, 0.15, 1.35], 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 = 9.1 kg and a = 12.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 109.655, "unit": "N"}

Answer: 109.655

Prompt: A cart has mass 9.1 kg and measured acceleration 12.05 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 = 9.1 kg and a = 12.05 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 32.05^2 and multiply by 0.5 * 13.85.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 7113.3773, "unit": "J"}

Answer: 7113.3773

Prompt: A body of mass 13.85 kg moves at 32.05 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 32.05^2 and multiply by 0.5 * 13.85.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 0.88 * 8.314 * 285 / 32.4.
Final: {"answer": 64.3565, "unit": "kPa"}

Answer: 64.3565

Prompt: An ideal gas sample has n=0.88 mol, T=285 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 = 0.88 * 8.314 * 285 / 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 = 1990 * 1.82 * 11.
Final: {"answer": 39839.8, "unit": "J"}

Answer: 39839.8

Prompt: A 1990 g sample has specific heat 1.82 J/(g*K). How much heat is needed to raise its temperature by 11 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1990 * 1.82 * 11.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=85.3 V and R=122.5 ohm.
Final: {"answer": 59.3967, "unit": "W"}

Answer: 59.3967

Prompt: A resistor has resistance 122.5 ohm and voltage 85.3 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=85.3 V and R=122.5 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=882.5 mg, t=283.5 h, and T_half=40.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 6.8945, "unit": "mg"}

Answer: 6.8945

Prompt: A radioactive sample starts with 882.5 mg. Its half-life is 40.5 hours. How many mg remain after 283.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=882.5 mg, t=283.5 h, and T_half=40.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 4.7 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 206.847, "unit": "g"}

Answer: 206.847

Prompt: A sample contains 4.7 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 4.7 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 4480 mL to 4.48 L.
3. Compute M = 4.405 / 4.48.
Final: {"answer": 0.9833, "unit": "mol/L"}

Answer: 0.9833

Prompt: A solution contains 4.405 mol of solute in 4480 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4480 mL to 4.48 L.
Compute M = 4.405 / 4.48.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.36 y, so each step multiplies y by 1 - h*0.36.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 1.4052, "unit": "state_units"}

Answer: 1.4052

Prompt: Use explicit Euler on dy/dt = -0.36 y with y(0)=13.5, step size h=1.2, for 4 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.36 y, so each step multiplies y by 1 - h*0.36.
Apply that update for 4 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-5)=32.4 and f(0.5)=0.5.
3. The interval width is 5.5.
Final: {"answer": 90.475, "unit": "area_units"}

Answer: 90.475

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-5.8x+3.4 from x=-5 to x=0.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-5)=32.4 and f(0.5)=0.5.
The interval width is 5.5.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 3357.5 / 100.
Final: {"answer": 33.575, "unit": "m"}

Answer: 33.575

Prompt: Convert 3357.5 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 3357.5 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=21 degrees and keep the unit as newtons.
Final: {"answer": [285.6756, 109.6606], "unit": "N"}

Answer: 285.6756

Answer: 109.6606

Prompt: A vector has magnitude 306 N at 21 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=21 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": [6.5, -10.75], "unit": "dimensionless"}

Answer: 6.5

Answer: -10.75

Prompt: A line passes through points (-9.75, -74.125) and (-3.5, -33.5). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: inconsistent

Prompt: In student derivation case 209886, 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_002500(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_002500"}

Answer: python_code

Prompt: Write a Python function unit_vector_002500(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_002500(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)=26.5 and h=1.55, 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. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 18.8^2 and multiply by 0.5 * 28.2.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 4983.504, "unit": "J"}

Answer: 4983.504

Prompt: A body of mass 28.2 kg moves at 18.8 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 18.8^2 and multiply by 0.5 * 28.2.
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=45 degrees through the sine of 2 theta.
3. Substitute v=66.7 m/s and g=9.81 m/s^2.
Final: {"answer": 453.5056, "unit": "m"}

Answer: 453.5056

Prompt: A projectile is launched on level ground with no air resistance at speed 66.7 m/s and angle 45 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=45 degrees through the sine of 2 theta.
Substitute v=66.7 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.81 * 8.314 * 359 / 21.8.
Final: {"answer": 795.4706, "unit": "kPa"}

Answer: 795.4706

Prompt: An ideal gas sample has n=5.81 mol, T=359 K, and V=21.8 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 = 5.81 * 8.314 * 359 / 21.8.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1760 * 1.87 * 62.
Final: {"answer": 204054.4, "unit": "J"}

Answer: 204054.4

Prompt: A 1760 g sample has specific heat 1.87 J/(g*K). How much heat is needed to raise its temperature by 62 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1760 * 1.87 * 62.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=222.4 V and R=116.5 ohm.
Final: {"answer": 424.5645, "unit": "W"}

Answer: 424.5645

Prompt: A resistor has resistance 116.5 ohm and voltage 222.4 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=222.4 V and R=116.5 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=602 mg, t=86.5 h, and T_half=53 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 194.2195, "unit": "mg"}

Answer: 194.2195

Prompt: A radioactive sample starts with 602 mg. Its half-life is 53 hours. How many mg remain after 86.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=602 mg, t=86.5 h, and T_half=53 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.94 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 129.3894, "unit": "g"}

Answer: 129.3894

Prompt: A sample contains 2.94 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 = 2.94 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 3000 mL to 3 L.
3. Compute M = 1.165 / 3.
Final: {"answer": 0.3883, "unit": "mol/L"}

Answer: 0.3883

Prompt: A solution contains 1.165 mol of solute in 3000 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3000 mL to 3 L.
Compute M = 1.165 / 3.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.34 y, so each step multiplies y by 1 - h*0.34.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 4.0119, "unit": "state_units"}

Answer: 4.0119

Prompt: Use explicit Euler on dy/dt = -0.34 y with y(0)=62, step size h=0.6, 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) = -0.34 y, so each step multiplies y by 1 - h*0.34.
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(2.4)=-19.68 and f(7.8)=-61.26.
3. The interval width is 5.4.
Final: {"answer": -218.538, "unit": "area_units"}

Answer: -218.538

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-7.7x+-1.2 from x=2.4 to x=7.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(2.4)=-19.68 and f(7.8)=-61.26.
The interval width is 5.4.


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=0.1 and add it to the current value.
Final: {"answer": [1.24, 1.0, 1.91], "unit": "temperature_units"}

Answer: 1.24

Answer: 1.0

Answer: 1.91

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=-0.35, right=-1.65; current interior values are [1.5, 0.75, 2.5]. Use r=0.1. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.1 and add it to the current value.


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 35801.0 / 1000.
Final: {"answer": 35.801, "unit": "kg"}

Answer: 35.801

Prompt: Convert 35801 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 35801.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=348 degrees and keep the unit as newtons.
Final: {"answer": [223.5067, -47.5078], "unit": "N"}

Answer: 223.5067

Answer: -47.5078

Prompt: A vector has magnitude 228.5 N at 348 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=348 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, -4.75], "unit": "dimensionless"}

Answer: 8.1

Answer: -4.75

Prompt: A line passes through points (-16.75, -140.425) and (-3, -29.05). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. density times acceleration times height gives kg/(m*s^2), or pascal
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In field worksheet case 647831, check dimensional consistency for p = rho * g * h. The left side has dimensions Pa and the right side has dimensions Pa. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
density times acceleration times height gives kg/(m*s^2), or pascal
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
def finite_differences_002518(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_002518"}

Answer: python_code

Prompt: Write a Python function finite_differences_002518(values) that returns the list of successive differences values[i+1] - values[i]. It must return [] for fewer than two values. Return a single fenced Python code block.

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

assert finite_differences_002518([5]) == []

assert finite_differences_002518([]) == []


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)=61.5 and h=1.25, 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 = 17.15 kg and a = 5.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 86.6075, "unit": "N"}

Answer: 86.6075

Prompt: A cart has mass 17.15 kg and measured acceleration 5.05 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 = 17.15 kg and a = 5.05 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 25.25^2 and multiply by 0.5 * 13.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 4223.8516, "unit": "J"}

Answer: 4223.8516

Prompt: A body of mass 13.25 kg moves at 25.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 25.25^2 and multiply by 0.5 * 13.25.
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=65 degrees through the sine of 2 theta.
3. Substitute v=16.9 m/s and g=9.81 m/s^2.
Final: {"answer": 22.3027, "unit": "m"}

Answer: 22.3027

Prompt: A projectile is launched on level ground with no air resistance at speed 16.9 m/s and angle 65 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=65 degrees through the sine of 2 theta.
Substitute v=16.9 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 * 8.314 * 616 / 88.1.
Final: {"answer": 232.5278, "unit": "kPa"}

Answer: 232.5278

Prompt: An ideal gas sample has n=4 mol, T=616 K, and V=88.1 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4 * 8.314 * 616 / 88.1.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1335 * 4.035 * 46.
Final: {"answer": 247789.35, "unit": "J"}

Answer: 247789.35

Prompt: A 1335 g sample has specific heat 4.035 J/(g*K). How much heat is needed to raise its temperature by 46 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1335 * 4.035 * 46.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=45.6 V and R=53.6 ohm.
Final: {"answer": 38.794, "unit": "W"}

Answer: 38.794

Prompt: A resistor has resistance 53.6 ohm and voltage 45.6 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=45.6 V and R=53.6 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=111 mg, t=357 h, and T_half=4.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0, "unit": "mg"}

Answer: 0.0

Prompt: A radioactive sample starts with 111 mg. Its half-life is 4.5 hours. How many mg remain after 357 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=111 mg, t=357 h, and T_half=4.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 5.56 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1001.6674, "unit": "g"}

Answer: 1001.6674

Prompt: A sample contains 5.56 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 5.56 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 890 mL to 0.89 L.
3. Compute M = 2.195 / 0.89.
Final: {"answer": 2.4663, "unit": "mol/L"}

Answer: 2.4663

Prompt: A solution contains 2.195 mol of solute in 890 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 890 mL to 0.89 L.
Compute M = 2.195 / 0.89.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-2.1)=14.26 and f(-0.5)=13.3.
3. The interval width is 1.6.
Final: {"answer": 22.048, "unit": "area_units"}

Answer: 22.048

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-0.6x+13 from x=-2.1 to x=-0.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-2.1)=14.26 and f(-0.5)=13.3.
The interval width is 1.6.


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": [-1.092, 0.672, -1.056], "unit": "temperature_units"}

Answer: -1.092

Answer: 0.672

Answer: -1.056

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=-1.3, right=-0.9; current interior values are [2.7, -1.2, -1.2]. 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. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 49912.0 / 1000.
Final: {"answer": 49.912, "unit": "kg"}

Answer: 49.912

Prompt: Convert 49912 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 49912.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=208 degrees and keep the unit as newtons.
Final: {"answer": [-114.7832, -61.0313], "unit": "N"}

Answer: -114.7832

Answer: -61.0313

Prompt: A vector has magnitude 130 N at 208 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=208 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-9.75, -5.25], "unit": "dimensionless"}

Answer: -9.75

Answer: -5.25

Prompt: A line passes through points (0.5, -10.125) and (8.25, -85.6875). 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 model card equation case 919623, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

```python
def finite_differences_002536(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_002536"}

Answer: python_code

Prompt: Write a Python function finite_differences_002536(values) that returns the list of successive differences values[i+1] - values[i]. It must return [] for fewer than two values. Return a single fenced Python code block.

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

assert finite_differences_002536([5]) == []

assert finite_differences_002536([]) == []


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 field sensor record from batch 728748 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 = 22.75 kg and a = 13.2 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 300.3, "unit": "N"}

Answer: 300.3

Prompt: A cart has mass 22.75 kg and measured acceleration 13.2 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 22.75 kg and a = 13.2 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 19^2 and multiply by 0.5 * 19.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 3474.625, "unit": "J"}

Answer: 3474.625

Prompt: A body of mass 19.25 kg moves at 19 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 19^2 and multiply by 0.5 * 19.25.
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=67 degrees through the sine of 2 theta.
3. Substitute v=14.2 m/s and g=9.81 m/s^2.
Final: {"answer": 14.7857, "unit": "m"}

Answer: 14.7857

Prompt: A projectile is launched on level ground with no air resistance at speed 14.2 m/s and angle 67 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=67 degrees through the sine of 2 theta.
Substitute v=14.2 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 = 4.23 * 8.314 * 305 / 110.4.
Final: {"answer": 97.1586, "unit": "kPa"}

Answer: 97.1586

Prompt: An ideal gas sample has n=4.23 mol, T=305 K, and V=110.4 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 = 4.23 * 8.314 * 305 / 110.4.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 645 * 3.81 * 120.
Final: {"answer": 294894.0, "unit": "J"}

Answer: 294894.0

Prompt: A 645 g sample has specific heat 3.81 J/(g*K). How much heat is needed to raise its temperature by 120 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 645 * 3.81 * 120.


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=211.9 V and R=104 ohm.
Final: {"answer": 431.7463, "unit": "W"}

Answer: 431.7463

Prompt: A resistor has resistance 104 ohm and voltage 211.9 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=211.9 V and R=104 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 4.01 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 401.3489, "unit": "g"}

Answer: 401.3489

Prompt: A sample contains 4.01 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 = 4.01 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 4240 mL to 4.24 L.
3. Compute M = 3.685 / 4.24.
Final: {"answer": 0.8691, "unit": "mol/L"}

Answer: 0.8691

Prompt: A solution contains 3.685 mol of solute in 4240 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4240 mL to 4.24 L.
Compute M = 3.685 / 4.24.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.6)=10.68 and f(-0.8)=-11.36.
3. The interval width is 3.8.
Final: {"answer": -1.292, "unit": "area_units"}

Answer: -1.292

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-5.8x+-16 from x=-4.6 to x=-0.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.6)=10.68 and f(-0.8)=-11.36.
The interval width is 3.8.


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=0.32 and add it to the current value.
Final: {"answer": [-0.53, -0.446, -0.624], "unit": "temperature_units"}

Answer: -0.53

Answer: -0.446

Answer: -0.624

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.85, right=0.95; current interior values are [1.55, -1.55, -1.2]. Use r=0.32. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.32 and add it to the current value.


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute -61.5 + 273.15.
Final: {"answer": 211.65, "unit": "K"}

Answer: 211.65

Prompt: Convert -61.5 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute -61.5 + 273.15.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=355 degrees and keep the unit as newtons.
Final: {"answer": [383.0369, -33.5114], "unit": "N"}

Answer: 383.0369

Answer: -33.5114

Prompt: A vector has magnitude 384.5 N at 355 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=355 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-9.95, -17.75], "unit": "dimensionless"}

Answer: -9.95

Answer: -17.75

Prompt: A line passes through points (-14.75, 129.012) and (-7.5, 56.875). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. 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 982364, check dimensional consistency for Q = m * c * DeltaT. The left side has dimensions J and the right side has dimensions J. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

```python
import math

def unit_vector_002554(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_002554"}

Answer: python_code

Prompt: Write a Python function unit_vector_002554(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_002554(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: left_boundary, right_boundary.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["left_boundary", "right_boundary"]}

Answer: INSUFFICIENT_INFO

Prompt: Solve the 1D heat equation after one finite-difference step. The prompt gives r=0.1 and interior values [1.3, 2.55, 0.85], 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 = 8.35 kg and a = 14.1 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 117.735, "unit": "N"}

Answer: 117.735

Prompt: A cart has mass 8.35 kg and measured acceleration 14.1 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 8.35 kg and a = 14.1 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=28 degrees through the sine of 2 theta.
3. Substitute v=59.9 m/s and g=9.81 m/s^2.
Final: {"answer": 303.2207, "unit": "m"}

Answer: 303.2207

Prompt: A projectile is launched on level ground with no air resistance at speed 59.9 m/s and angle 28 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=28 degrees through the sine of 2 theta.
Substitute v=59.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.09 * 8.314 * 533 / 33.3.
Final: {"answer": 411.1985, "unit": "kPa"}

Answer: 411.1985

Prompt: An ideal gas sample has n=3.09 mol, T=533 K, and V=33.3 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 3.09 * 8.314 * 533 / 33.3.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1795 * 4.25 * 65.
Final: {"answer": 495868.75, "unit": "J"}

Answer: 495868.75

Prompt: A 1795 g sample has specific heat 4.25 J/(g*K). How much heat is needed to raise its temperature by 65 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1795 * 4.25 * 65.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=818.5 mg, t=255 h, and T_half=2 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0, "unit": "mg"}

Answer: 0.0

Prompt: A radioactive sample starts with 818.5 mg. Its half-life is 2 hours. How many mg remain after 255 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=818.5 mg, t=255 h, and T_half=2 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 5.09 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 509.4428, "unit": "g"}

Answer: 509.4428

Prompt: A sample contains 5.09 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 = 5.09 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 2215 mL to 2.215 L.
3. Compute M = 0.755 / 2.215.
Final: {"answer": 0.3409, "unit": "mol/L"}

Answer: 0.3409

Prompt: A solution contains 0.755 mol of solute in 2215 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2215 mL to 2.215 L.
Compute M = 0.755 / 2.215.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.06 and add it to the current value.
Final: {"answer": [0.97, 0.052, -1.42], "unit": "temperature_units"}

Answer: 0.97

Answer: 0.052

Answer: -1.42

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-0.8, right=1.9; current interior values are [1.15, 0.1, -1.75]. 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. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 186.5 + 273.15.
Final: {"answer": 459.65, "unit": "K"}

Answer: 459.65

Prompt: Convert 186.5 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute 186.5 + 273.15.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=177 degrees and keep the unit as newtons.
Final: {"answer": [-205.2184, 10.755], "unit": "N"}

Answer: -205.2184

Answer: 10.755

Prompt: A vector has magnitude 205.5 N at 177 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=177 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": [-8.45, -27.0], "unit": "dimensionless"}

Answer: -8.45

Answer: -27.0

Prompt: A line passes through points (-11.5, 70.175) and (-3.75, 4.6875). Return [slope, intercept] for y = slope*x + intercept.

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


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

```python
def pairwise_ratios_002572(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_002572"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_002572(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_002572([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_002572([1, 2], [1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=52.5 and h=0.55, 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 = 21.05 kg and a = 12.65 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 266.2825, "unit": "N"}

Answer: 266.2825

Prompt: A cart has mass 21.05 kg and measured acceleration 12.65 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 = 21.05 kg and a = 12.65 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 = 23.4^2 and multiply by 0.5 * 19.5.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 5338.71, "unit": "J"}

Answer: 5338.71

Prompt: A body of mass 19.5 kg moves at 23.4 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 23.4^2 and multiply by 0.5 * 19.5.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=15 degrees through the sine of 2 theta.
3. Substitute v=76.8 m/s and g=9.81 m/s^2.
Final: {"answer": 300.6239, "unit": "m"}

Answer: 300.6239

Prompt: A projectile is launched on level ground with no air resistance at speed 76.8 m/s and angle 15 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=15 degrees through the sine of 2 theta.
Substitute v=76.8 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 = 2.76 * 8.314 * 616 / 117.2.
Final: {"answer": 120.6069, "unit": "kPa"}

Answer: 120.6069

Prompt: An ideal gas sample has n=2.76 mol, T=616 K, and V=117.2 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2.76 * 8.314 * 616 / 117.2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 480 * 2.81 * 88.
Final: {"answer": 118694.4, "unit": "J"}

Answer: 118694.4

Prompt: A 480 g sample has specific heat 2.81 J/(g*K). How much heat is needed to raise its temperature by 88 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 480 * 2.81 * 88.


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=224.2 V and R=122.7 ohm.
Final: {"answer": 409.6629, "unit": "W"}

Answer: 409.6629

Prompt: A resistor has resistance 122.7 ohm and voltage 224.2 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=224.2 V and R=122.7 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=396.5 mg, t=64.5 h, and T_half=1.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0, "unit": "mg"}

Answer: 0.0

Prompt: A radioactive sample starts with 396.5 mg. Its half-life is 1.5 hours. How many mg remain after 64.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=396.5 mg, t=64.5 h, and T_half=1.5 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 = 3.73 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 373.3245, "unit": "g"}

Answer: 373.3245

Prompt: A sample contains 3.73 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 = 3.73 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) = -0.55 y, so each step multiplies y by 1 - h*0.55.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 0.0014, "unit": "state_units"}

Answer: 0.0014

Prompt: Use explicit Euler on dy/dt = -0.55 y with y(0)=78.5, step size h=1.7, for 4 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.55 y, so each step multiplies y by 1 - h*0.55.
Apply that update for 4 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-6)=-22.3 and f(-0.6)=-15.82.
3. The interval width is 5.4.
Final: {"answer": -102.924, "unit": "area_units"}

Answer: -102.924

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=1.2x+-15.1 from x=-6 to x=-0.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-6)=-22.3 and f(-0.6)=-15.82.
The interval width is 5.4.


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=0.38 and add it to the current value.
Final: {"answer": [1.513, 0.247, 1.243], "unit": "temperature_units"}

Answer: 1.513

Answer: 0.247

Answer: 1.243

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.45, right=2.35; current interior values are [1.0, 1.9, -1.55]. 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=58 degrees and keep the unit as newtons.
Final: {"answer": [100.9496, 161.5532], "unit": "N"}

Answer: 100.9496

Answer: 161.5532

Prompt: A vector has magnitude 190.5 N at 58 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=58 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [4.9, 28.5], "unit": "dimensionless"}

Answer: 4.9

Answer: 28.5

Prompt: A line passes through points (13.25, 93.425) and (16.5, 109.35). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: consistent

Prompt: In field worksheet case 991512, 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 pairwise_ratios_002590(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_002590"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_002590(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_002590([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_002590([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=1.37 mol and T=323 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. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=67 degrees through the sine of 2 theta.
3. Substitute v=7.4 m/s and g=9.81 m/s^2.
Final: {"answer": 4.0154, "unit": "m"}

Answer: 4.0154

Prompt: A projectile is launched on level ground with no air resistance at speed 7.4 m/s and angle 67 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=67 degrees through the sine of 2 theta.
Substitute v=7.4 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 = 0.78 * 8.314 * 260 / 27.3.
Final: {"answer": 61.7611, "unit": "kPa"}

Answer: 61.7611

Prompt: An ideal gas sample has n=0.78 mol, T=260 K, and V=27.3 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 0.78 * 8.314 * 260 / 27.3.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=233.5 mg, t=46 h, and T_half=38 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 100.8979, "unit": "mg"}

Answer: 100.8979

Prompt: A radioactive sample starts with 233.5 mg. Its half-life is 38 hours. How many mg remain after 46 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=233.5 mg, t=46 h, and T_half=38 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 = 3.14 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 56.5671, "unit": "g"}

Answer: 56.5671

Prompt: A sample contains 3.14 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 = 3.14 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2120 mL to 2.12 L.
3. Compute M = 0.955 / 2.12.
Final: {"answer": 0.4505, "unit": "mol/L"}

Answer: 0.4505

Prompt: A solution contains 0.955 mol of solute in 2120 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2120 mL to 2.12 L.
Compute M = 0.955 / 2.12.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.32 y, so each step multiplies y by 1 - h*0.32.
3. Apply that update for 10 steps and round the final state.
Final: {"answer": 8.8968, "unit": "state_units"}

Answer: 8.8968

Prompt: Use explicit Euler on dy/dt = -0.32 y with y(0)=35, step size h=0.4, for 10 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.32 y, so each step multiplies y by 1 - h*0.32.
Apply that update for 10 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-1.5)=-2.9 and f(8.5)=-62.9.
3. The interval width is 10.
Final: {"answer": -329.0, "unit": "area_units"}

Answer: -329.0

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-6x+-11.9 from x=-1.5 to x=8.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-1.5)=-2.9 and f(8.5)=-62.9.
The interval width is 10.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.28 and add it to the current value.
Final: {"answer": [-0.546, -0.07, 0.596], "unit": "temperature_units"}

Answer: -0.546

Answer: -0.07

Answer: 0.596

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.55, right=2.45; current interior values are [-0.7, 0.7, -0.65]. Use r=0.28. Return the next interior vector.

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


Response:
Reasoning:
1. 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": [184.5646, -254.0313], "unit": "N"}

Answer: 184.5646

Answer: -254.0313

Prompt: A vector has magnitude 314 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. 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 student derivation case 596251, 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 clamp_002608(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_002608"}

Answer: python_code

Prompt: Write a Python function clamp_002608(x, lo, hi) that clips x into the inclusive interval [lo, hi]. It must raise ValueError if lo > hi. Return a single fenced Python code block.

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

try:
    clamp_002608(1, 3, 2)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a segmentation scoring problem with 2249 items and budget 96735, 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 = 38.55^2 and multiply by 0.5 * 15.05.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 11182.9213, "unit": "J"}

Answer: 11182.9213

Prompt: A body of mass 15.05 kg moves at 38.55 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 38.55^2 and multiply by 0.5 * 15.05.
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=65 degrees through the sine of 2 theta.
3. Substitute v=61.7 m/s and g=9.81 m/s^2.
Final: {"answer": 297.2729, "unit": "m"}

Answer: 297.2729

Prompt: A projectile is launched on level ground with no air resistance at speed 61.7 m/s and angle 65 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=65 degrees through the sine of 2 theta.
Substitute v=61.7 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 = 1580 * 3.4 * 51.
Final: {"answer": 273972.0, "unit": "J"}

Answer: 273972.0

Prompt: A 1580 g sample has specific heat 3.4 J/(g*K). How much heat is needed to raise its temperature by 51 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1580 * 3.4 * 51.


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=195.1 V and R=127.3 ohm.
Final: {"answer": 299.0103, "unit": "W"}

Answer: 299.0103

Prompt: A resistor has resistance 127.3 ohm and voltage 195.1 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=195.1 V and R=127.3 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=943.5 mg, t=203.5 h, and T_half=38 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 23.0487, "unit": "mg"}

Answer: 23.0487

Prompt: A radioactive sample starts with 943.5 mg. Its half-life is 38 hours. How many mg remain after 203.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=943.5 mg, t=203.5 h, and T_half=38 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 = 6.83 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 123.0425, "unit": "g"}

Answer: 123.0425

Prompt: A sample contains 6.83 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.83 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 735 mL to 0.735 L.
3. Compute M = 2.945 / 0.735.
Final: {"answer": 4.0068, "unit": "mol/L"}

Answer: 4.0068

Prompt: A solution contains 2.945 mol of solute in 735 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 735 mL to 0.735 L.
Compute M = 2.945 / 0.735.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-8.5)=-54.2 and f(-3.9)=-26.6.
3. The interval width is 4.6.
Final: {"answer": -185.84, "unit": "area_units"}

Answer: -185.84

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=6x+-3.2 from x=-8.5 to x=-3.9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-8.5)=-54.2 and f(-3.9)=-26.6.
The interval width is 4.6.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.28 and add it to the current value.
Final: {"answer": [-0.228, 0.794, 1.708], "unit": "temperature_units"}

Answer: -0.228

Answer: 0.794

Answer: 1.708

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=2.65; current interior values are [0.5, 0.15, 2.1]. Use r=0.28. Return the next interior vector.

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


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 3206.0 / 1000.
Final: {"answer": 3.206, "unit": "kg"}

Answer: 3.206

Prompt: Convert 3206 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 3206.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=131 degrees and keep the unit as newtons.
Final: {"answer": [-295.8826, 340.374], "unit": "N"}

Answer: -295.8826

Answer: 340.374

Prompt: A vector has magnitude 451 N at 131 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=131 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. density times acceleration times height gives kg/(m*s^2), or pascal
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In field worksheet case 490565, check dimensional consistency for p = rho * g * h. The left side has dimensions Pa and the right side has dimensions Pa. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
density times acceleration times height gives kg/(m*s^2), or pascal
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. 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)=58 and h=1.7, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 19.45 kg and a = 4.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 83.635, "unit": "N"}

Answer: 83.635

Prompt: A cart has mass 19.45 kg and measured acceleration 4.3 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 19.45 kg and a = 4.3 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=59 degrees through the sine of 2 theta.
3. Substitute v=81.5 m/s and g=9.81 m/s^2.
Final: {"answer": 597.8347, "unit": "m"}

Answer: 597.8347

Prompt: A projectile is launched on level ground with no air resistance at speed 81.5 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=81.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 1.72 * 8.314 * 348 / 50.4.
Final: {"answer": 98.7386, "unit": "kPa"}

Answer: 98.7386

Prompt: An ideal gas sample has n=1.72 mol, T=348 K, and V=50.4 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 = 1.72 * 8.314 * 348 / 50.4.


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=55.3 V and R=1.9 ohm.
Final: {"answer": 1609.5211, "unit": "W"}

Answer: 1609.5211

Prompt: A resistor has resistance 1.9 ohm and voltage 55.3 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=55.3 V and R=1.9 ohm.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4780 mL to 4.78 L.
3. Compute M = 3.54 / 4.78.
Final: {"answer": 0.7406, "unit": "mol/L"}

Answer: 0.7406

Prompt: A solution contains 3.54 mol of solute in 4780 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4780 mL to 4.78 L.
Compute M = 3.54 / 4.78.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.99 y, so each step multiplies y by 1 - h*0.99.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": -11.5395, "unit": "state_units"}

Answer: -11.5395

Prompt: Use explicit Euler on dy/dt = -0.99 y with y(0)=46.5, step size h=1.9, 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) = -0.99 y, so each step multiplies y by 1 - h*0.99.
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(1.8)=-7.4 and f(11.1)=-77.15.
3. The interval width is 9.3.
Final: {"answer": -393.1575, "unit": "area_units"}

Answer: -393.1575

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-7.5x+6.1 from x=1.8 to x=11.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(1.8)=-7.4 and f(11.1)=-77.15.
The interval width is 9.3.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.38 and add it to the current value.
Final: {"answer": [1.139, 0.933, 1.435], "unit": "temperature_units"}

Answer: 1.139

Answer: 0.933

Answer: 1.435

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.25, right=2.15; current interior values are [2.45, 2.7, -1.7]. 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. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 7770.0 / 1000.
Final: {"answer": 7.77, "unit": "kg"}

Answer: 7.77

Prompt: Convert 7770 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 7770.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=322 degrees and keep the unit as newtons.
Final: {"answer": [306.5362, -239.4923], "unit": "N"}

Answer: 306.5362

Answer: -239.4923

Prompt: A vector has magnitude 389 N at 322 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=322 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": [-1.45, 29.75], "unit": "dimensionless"}

Answer: -1.45

Answer: 29.75

Prompt: A line passes through points (-7.75, 40.9875) and (2, 26.85). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: consistent

Prompt: In quality-control check case 662902, check dimensional consistency for Q = m * c * DeltaT. The left side has dimensions J and the right side has dimensions J. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

```python
def finite_differences_002644(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_002644"}

Answer: python_code

Prompt: Write a Python function finite_differences_002644(values) that returns the list of successive differences values[i+1] - values[i]. It must return [] for fewer than two values. Return a single fenced Python code block.

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

assert finite_differences_002644([5]) == []

assert finite_differences_002644([]) == []


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)=11.5 and h=1.45, 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 = 1.2 kg and a = 15.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 18.06, "unit": "N"}

Answer: 18.06

Prompt: A cart has mass 1.2 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 = 1.2 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=38 degrees through the sine of 2 theta.
3. Substitute v=31.6 m/s and g=9.81 m/s^2.
Final: {"answer": 98.7664, "unit": "m"}

Answer: 98.7664

Prompt: A projectile is launched on level ground with no air resistance at speed 31.6 m/s and angle 38 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=38 degrees through the sine of 2 theta.
Substitute v=31.6 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 = 1540 * 3.98 * 108.
Final: {"answer": 661953.6, "unit": "J"}

Answer: 661953.6

Prompt: A 1540 g sample has specific heat 3.98 J/(g*K). How much heat is needed to raise its temperature by 108 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1540 * 3.98 * 108.


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=228.8 V and R=113.2 ohm.
Final: {"answer": 462.4509, "unit": "W"}

Answer: 462.4509

Prompt: A resistor has resistance 113.2 ohm and voltage 228.8 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=228.8 V and R=113.2 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=72 mg, t=199 h, and T_half=54 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 5.5973, "unit": "mg"}

Answer: 5.5973

Prompt: A radioactive sample starts with 72 mg. Its half-life is 54 hours. How many mg remain after 199 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=72 mg, t=199 h, and T_half=54 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.05 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 90.2205, "unit": "g"}

Answer: 90.2205

Prompt: A sample contains 2.05 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 = 2.05 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 1330 mL to 1.33 L.
3. Compute M = 3.25 / 1.33.
Final: {"answer": 2.4436, "unit": "mol/L"}

Answer: 2.4436

Prompt: A solution contains 3.25 mol of solute in 1330 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1330 mL to 1.33 L.
Compute M = 3.25 / 1.33.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.18 y, so each step multiplies y by 1 - h*0.18.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 3.6722, "unit": "state_units"}

Answer: 3.6722

Prompt: Use explicit Euler on dy/dt = -0.18 y with y(0)=78.5, step size h=1.35, 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) = -0.18 y, so each step multiplies y by 1 - h*0.18.
Apply that update for 11 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(0.3)=5.09 and f(9.8)=17.44.
3. The interval width is 9.5.
Final: {"answer": 107.0175, "unit": "area_units"}

Answer: 107.0175

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=1.3x+4.7 from x=0.3 to x=9.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(0.3)=5.09 and f(9.8)=17.44.
The interval width is 9.5.


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=0.32 and add it to the current value.
Final: {"answer": [-0.234, -0.43, -0.314], "unit": "temperature_units"}

Answer: -0.234

Answer: -0.43

Answer: -0.314

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=-1.05, right=1.4; current interior values are [0.95, -0.75, -1.45]. Use r=0.32. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.32 and add it to the current value.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=191 degrees and keep the unit as newtons.
Final: {"answer": [-45.1549, -8.7772], "unit": "N"}

Answer: -45.1549

Answer: -8.7772

Prompt: A vector has magnitude 46 N at 191 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=191 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [4.75, 4.25], "unit": "dimensionless"}

Answer: 4.75

Answer: 4.25

Prompt: A line passes through points (13.75, 69.5625) and (15, 75.5). Return [slope, intercept] for y = slope*x + intercept.

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


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

Answer: inconsistent

Prompt: In lab notebook case 978735, 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 finite_differences_002662(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_002662"}

Answer: python_code

Prompt: Write a Python function finite_differences_002662(values) that returns the list of successive differences values[i+1] - values[i]. It must return [] for fewer than two values. Return a single fenced Python code block.

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

assert finite_differences_002662([5]) == []

assert finite_differences_002662([]) == []


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 2585 items and budget 22185, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 18.75 kg and a = 12.1 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 226.875, "unit": "N"}

Answer: 226.875

Prompt: A cart has mass 18.75 kg and measured acceleration 12.1 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 18.75 kg and a = 12.1 m/s^2.
The unit kg*m/s^2 is one newton.