Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 10.15^2 and multiply by 0.5 * 26.95.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1388.2282, "unit": "J"}

Answer: 1388.2282

Prompt: A body of mass 26.95 kg moves at 10.15 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 10.15^2 and multiply by 0.5 * 26.95.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=64 degrees through the sine of 2 theta.
3. Substitute v=46.7 m/s and g=9.81 m/s^2.
Final: {"answer": 175.185, "unit": "m"}

Answer: 175.185

Prompt: A projectile is launched on level ground with no air resistance at speed 46.7 m/s and angle 64 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=64 degrees through the sine of 2 theta.
Substitute v=46.7 m/s and g=9.81 m/s^2.

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


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.94 * 8.314 * 449 / 10.4.
Final: {"answer": 1773.1684, "unit": "kPa"}

Answer: 1773.1684

Prompt: An ideal gas sample has n=4.94 mol, T=449 K, and V=10.4 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 = 4.94 * 8.314 * 449 / 10.4.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=768.5 mg, t=282 h, and T_half=60 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 29.5667, "unit": "mg"}

Answer: 29.5667

Prompt: A radioactive sample starts with 768.5 mg. Its half-life is 60 hours. How many mg remain after 282 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=768.5 mg, t=282 h, and T_half=60 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 = 6.4 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 374.016, "unit": "g"}

Answer: 374.016

Prompt: A sample contains 6.4 mol of sodium chloride. Using molar mass 58.44 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.4 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4455 mL to 4.455 L.
3. Compute M = 2.805 / 4.455.
Final: {"answer": 0.6296, "unit": "mol/L"}

Answer: 0.6296

Prompt: A solution contains 2.805 mol of solute in 4455 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4455 mL to 4.455 L.
Compute M = 2.805 / 4.455.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.66 y, so each step multiplies y by 1 - h*0.66.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 0.0, "unit": "state_units"}

Answer: 0.0

Prompt: Use explicit Euler on dy/dt = -0.66 y with y(0)=67, step size h=1.1, 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) = -0.66 y, so each step multiplies y by 1 - h*0.66.
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(6.1)=-10.62 and f(13.7)=-19.74.
3. The interval width is 7.6.
Final: {"answer": -115.368, "unit": "area_units"}

Answer: -115.368

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-1.2x+-3.3 from x=6.1 to x=13.7. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(6.1)=-10.62 and f(13.7)=-19.74.
The interval width is 7.6.

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


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.12 and add it to the current value.
Final: {"answer": [-0.314, 1.59, -0.828], "unit": "temperature_units"}

Answer: -0.314

Answer: 1.59

Answer: -0.828

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=0.85, right=2.9; current interior values are [-0.95, 2.55, -1.95]. Use r=0.12. Return the next interior vector.

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


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 29100.0 / 1000.
Final: {"answer": 29.1, "unit": "kg"}

Answer: 29.1

Prompt: Convert 29100 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 29100.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=151 degrees and keep the unit as newtons.
Final: {"answer": [-38.4833, 21.3316], "unit": "N"}

Answer: -38.4833

Answer: 21.3316

Prompt: A vector has magnitude 44 N at 151 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=151 degrees and keep the unit as newtons.

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


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

Answer: consistent

Prompt: In lab notebook case 524258, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

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

Answer: consistent


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

```python
def threshold_flags_000250(values, threshold):
    return [value >= threshold for value in values]
```
Final: {"answer": "python_code", "function": "threshold_flags_000250"}

Answer: python_code

Prompt: Write a Python function threshold_flags_000250(values, threshold) that returns a list of booleans showing whether each value is greater than or equal to threshold. Return a single fenced Python code block.

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

assert threshold_flags_000250([1, 2, 3], 2) == [False, True, True]

assert threshold_flags_000250([], 5) == []

assert threshold_flags_000250([-1, 0], 0) == [False, True]


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: 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=0.3 and interior values [-1.05, -0.7, 0.1], but gives no boundary values. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 1.8 kg and a = 12.9 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 23.22, "unit": "N"}

Answer: 23.22

Prompt: A cart has mass 1.8 kg and measured acceleration 12.9 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 1.8 kg and a = 12.9 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 6.9^2 and multiply by 0.5 * 12.8.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 304.704, "unit": "J"}

Answer: 304.704

Prompt: A body of mass 12.8 kg moves at 6.9 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 6.9^2 and multiply by 0.5 * 12.8.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=23 degrees through the sine of 2 theta.
3. Substitute v=41.2 m/s and g=9.81 m/s^2.
Final: {"answer": 124.4685, "unit": "m"}

Answer: 124.4685

Prompt: A projectile is launched on level ground with no air resistance at speed 41.2 m/s and angle 23 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=23 degrees through the sine of 2 theta.
Substitute v=41.2 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 = 4.19 * 8.314 * 350 / 58.8.
Final: {"answer": 207.3551, "unit": "kPa"}

Answer: 207.3551

Prompt: An ideal gas sample has n=4.19 mol, T=350 K, and V=58.8 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.19 * 8.314 * 350 / 58.8.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=7 V and R=124 ohm.
Final: {"answer": 0.3952, "unit": "W"}

Answer: 0.3952

Prompt: A resistor has resistance 124 ohm and voltage 7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=7 V and R=124 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=856 mg, t=295.5 h, and T_half=40.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 5.4459, "unit": "mg"}

Answer: 5.4459

Prompt: A radioactive sample starts with 856 mg. Its half-life is 40.5 hours. How many mg remain after 295.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=856 mg, t=295.5 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 = 4.36 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 436.3793, "unit": "g"}

Answer: 436.3793

Prompt: A sample contains 4.36 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 = 4.36 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 1380 mL to 1.38 L.
3. Compute M = 3.185 / 1.38.
Final: {"answer": 2.308, "unit": "mol/L"}

Answer: 2.308

Prompt: A solution contains 3.185 mol of solute in 1380 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1380 mL to 1.38 L.
Compute M = 3.185 / 1.38.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.94 y, so each step multiplies y by 1 - h*0.94.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": -24.0132, "unit": "state_units"}

Answer: -24.0132

Prompt: Use explicit Euler on dy/dt = -0.94 y with y(0)=59.5, step size h=1.85, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.94 y, so each step multiplies y by 1 - h*0.94.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(4)=29.7 and f(8.1)=56.35.
3. The interval width is 4.1.
Final: {"answer": 176.4025, "unit": "area_units"}

Answer: 176.4025

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=6.5x+3.7 from x=4 to x=8.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(4)=29.7 and f(8.1)=56.35.
The interval width is 4.1.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.34 and add it to the current value.
Final: {"answer": [0.76, 0.62, 0.776], "unit": "temperature_units"}

Answer: 0.76

Answer: 0.62

Answer: 0.776

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=-0.05, right=2.35; current interior values are [1.95, 0.45, -0.55]. Use r=0.34. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.34 and add it to the current value.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 3459.5 / 100.
Final: {"answer": 34.595, "unit": "m"}

Answer: 34.595

Prompt: Convert 3459.5 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 3459.5 / 100.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-8.3, -3.5], "unit": "dimensionless"}

Answer: -8.3

Answer: -3.5

Prompt: A line passes through points (-2.25, 15.175) and (3.75, -34.625). Return [slope, intercept] for y = slope*x + intercept.

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


Response:
Reasoning:
1. 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 model card equation case 27109, check dimensional consistency for E = m * v. The left side has dimensions kg*m^2/s^2 and the right side has dimensions kg*m/s. Return consistent or inconsistent with a brief reason.

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

Answer: inconsistent


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

Answer: INSUFFICIENT_INFO

Prompt: Write a sanitizer for valid billing invoice records in release 979709, but the prompt does not state which fields are required or which transformations are allowed. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 16.95 kg and a = 11.45 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 194.0775, "unit": "N"}

Answer: 194.0775

Prompt: A cart has mass 16.95 kg and measured acceleration 11.45 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 16.95 kg and a = 11.45 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 25.1^2 and multiply by 0.5 * 29.9.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 9418.6495, "unit": "J"}

Answer: 9418.6495

Prompt: A body of mass 29.9 kg moves at 25.1 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 25.1^2 and multiply by 0.5 * 29.9.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=71 degrees through the sine of 2 theta.
3. Substitute v=61.6 m/s and g=9.81 m/s^2.
Final: {"answer": 238.1411, "unit": "m"}

Answer: 238.1411

Prompt: A projectile is launched on level ground with no air resistance at speed 61.6 m/s and angle 71 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=71 degrees through the sine of 2 theta.
Substitute v=61.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 0.51 * 8.314 * 392 / 111.5.
Final: {"answer": 14.907, "unit": "kPa"}

Answer: 14.907

Prompt: An ideal gas sample has n=0.51 mol, T=392 K, and V=111.5 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 0.51 * 8.314 * 392 / 111.5.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1300 * 0.565 * 95.
Final: {"answer": 69777.5, "unit": "J"}

Answer: 69777.5

Prompt: A 1300 g sample has specific heat 0.565 J/(g*K). How much heat is needed to raise its temperature by 95 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1300 * 0.565 * 95.


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=128.2 V and R=44.7 ohm.
Final: {"answer": 367.6787, "unit": "W"}

Answer: 367.6787

Prompt: A resistor has resistance 44.7 ohm and voltage 128.2 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=128.2 V and R=44.7 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=108 mg, t=356.5 h, and T_half=18.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0002, "unit": "mg"}

Answer: 0.0002

Prompt: A radioactive sample starts with 108 mg. Its half-life is 18.5 hours. How many mg remain after 356.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=108 mg, t=356.5 h, and T_half=18.5 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 = 3.81 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 167.6781, "unit": "g"}

Answer: 167.6781

Prompt: A sample contains 3.81 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 = 3.81 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 235 mL to 0.235 L.
3. Compute M = 0.06 / 0.235.
Final: {"answer": 0.2553, "unit": "mol/L"}

Answer: 0.2553

Prompt: A solution contains 0.06 mol of solute in 235 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 235 mL to 0.235 L.
Compute M = 0.06 / 0.235.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.69 y, so each step multiplies y by 1 - h*0.69.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 0.5891, "unit": "state_units"}

Answer: 0.5891

Prompt: Use explicit Euler on dy/dt = -0.69 y with y(0)=3.5, step size h=0.2, 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) = -0.69 y, so each step multiplies y by 1 - h*0.69.
Apply that update for 12 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-9.7)=-9.75 and f(-4.9)=-2.55.
3. The interval width is 4.8.
Final: {"answer": -29.52, "unit": "area_units"}

Answer: -29.52

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=1.5x+4.8 from x=-9.7 to x=-4.9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-9.7)=-9.75 and f(-4.9)=-2.55.
The interval width is 4.8.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.36 and add it to the current value.
Final: {"answer": [0.148, 0.614, 0.566], "unit": "temperature_units"}

Answer: 0.148

Answer: 0.614

Answer: 0.566

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.6, right=1.35; current interior values are [1.75, 0.65, -0.55]. Use r=0.36. 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=0.36 and add it to the current value.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 664.0 / 100.
Final: {"answer": 6.64, "unit": "m"}

Answer: 6.64

Prompt: Convert 664 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 664.0 / 100.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-5.45, -2.25], "unit": "dimensionless"}

Answer: -5.45

Answer: -2.25

Prompt: A line passes through points (-20, 106.75) and (-11, 57.7). 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 weighted_sum_000286(values, weights):
    if len(values) != len(weights):
        raise ValueError('length mismatch')
    return sum(value * weight for value, weight in zip(values, weights))
```
Final: {"answer": "python_code", "function": "weighted_sum_000286"}

Answer: python_code

Prompt: Write a Python function weighted_sum_000286(values, weights) that returns sum(value*weight). Raise ValueError when the two inputs have different lengths. Return a single fenced Python code block.

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

try:
    weighted_sum_000286([1], [1, 2])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=20.5 and h=0.15, 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 = 14.6 kg and a = 4.3 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 62.78, "unit": "N"}

Answer: 62.78

Prompt: A cart has mass 14.6 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 = 14.6 kg and a = 4.3 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 2.34 * 8.314 * 412 / 96.8.
Final: {"answer": 82.8033, "unit": "kPa"}

Answer: 82.8033

Prompt: An ideal gas sample has n=2.34 mol, T=412 K, and V=96.8 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2.34 * 8.314 * 412 / 96.8.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1490 * 1.8 * 6.
Final: {"answer": 16092.0, "unit": "J"}

Answer: 16092.0

Prompt: A 1490 g sample has specific heat 1.8 J/(g*K). How much heat is needed to raise its temperature by 6 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1490 * 1.8 * 6.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=71.7 V and R=112.6 ohm.
Final: {"answer": 45.6562, "unit": "W"}

Answer: 45.6562

Prompt: A resistor has resistance 112.6 ohm and voltage 71.7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=71.7 V and R=112.6 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=304.5 mg, t=150.5 h, and T_half=60 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 53.5185, "unit": "mg"}

Answer: 53.5185

Prompt: A radioactive sample starts with 304.5 mg. Its half-life is 60 hours. How many mg remain after 150.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=304.5 mg, t=150.5 h, and T_half=60 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 = 6.86 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 301.9086, "unit": "g"}

Answer: 301.9086

Prompt: A sample contains 6.86 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.86 mol * 44.01 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) = -0.58 y, so each step multiplies y by 1 - h*0.58.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": 0.9047, "unit": "state_units"}

Answer: 0.9047

Prompt: Use explicit Euler on dy/dt = -0.58 y with y(0)=24.5, step size h=1.15, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.58 y, so each step multiplies y by 1 - h*0.58.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-5.2)=-43.44 and f(3.5)=14.85.
3. The interval width is 8.7.
Final: {"answer": -124.3665, "unit": "area_units"}

Answer: -124.3665

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=6.7x+-8.6 from x=-5.2 to x=3.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-5.2)=-43.44 and f(3.5)=14.85.
The interval width is 8.7.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.3 and add it to the current value.
Final: {"answer": [0.09, 0.505, 0.715], "unit": "temperature_units"}

Answer: 0.09

Answer: 0.505

Answer: 0.715

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=-1.5, right=1.85; current interior values are [1.05, 0.4, 0.1]. Use r=0.3. Return the next interior vector.

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


Response:
Reasoning:
1. There are 1000 grams in one kilogram.
2. Apply the conversion factor directly to the given value.
3. Compute 22119.0 / 1000.
Final: {"answer": 22.119, "unit": "kg"}

Answer: 22.119

Prompt: Convert 22119 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 22119.0 / 1000.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=148 degrees and keep the unit as newtons.
Final: {"answer": [-355.7562, 222.3011], "unit": "N"}

Answer: -355.7562

Answer: 222.3011

Prompt: A vector has magnitude 419.5 N at 148 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=148 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.8, -24.75], "unit": "dimensionless"}

Answer: 8.8

Answer: -24.75

Prompt: A line passes through points (-7.25, -88.55) and (-4.5, -64.35). 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. velocity divided by time is acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In quality-control check case 655693, check dimensional consistency for x = v / t. The left side has dimensions m and the right side has dimensions m/s^2. Return consistent or inconsistent with a brief reason.

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

Answer: inconsistent


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

```python
def pairwise_ratios_000304(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_000304"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_000304(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_000304([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_000304([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=5.97 mol and T=347 K, but does not give volume. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 10.6^2 and multiply by 0.5 * 20.65.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1160.117, "unit": "J"}

Answer: 1160.117

Prompt: A body of mass 20.65 kg moves at 10.6 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 10.6^2 and multiply by 0.5 * 20.65.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=59 degrees through the sine of 2 theta.
3. Substitute v=28.5 m/s and g=9.81 m/s^2.
Final: {"answer": 73.1064, "unit": "m"}

Answer: 73.1064

Prompt: A projectile is launched on level ground with no air resistance at speed 28.5 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=28.5 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 = 5.9 * 8.314 * 444 / 74.5.
Final: {"answer": 292.3403, "unit": "kPa"}

Answer: 292.3403

Prompt: An ideal gas sample has n=5.9 mol, T=444 K, and V=74.5 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 5.9 * 8.314 * 444 / 74.5.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 455 * 4.665 * 74.
Final: {"answer": 157070.55, "unit": "J"}

Answer: 157070.55

Prompt: A 455 g sample has specific heat 4.665 J/(g*K). How much heat is needed to raise its temperature by 74 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 455 * 4.665 * 74.


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=50.7 V and R=150.1 ohm.
Final: {"answer": 17.1252, "unit": "W"}

Answer: 17.1252

Prompt: A resistor has resistance 150.1 ohm and voltage 50.7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=50.7 V and R=150.1 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=240.5 mg, t=10.5 h, and T_half=78 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 219.0745, "unit": "mg"}

Answer: 219.0745

Prompt: A radioactive sample starts with 240.5 mg. Its half-life is 78 hours. How many mg remain after 10.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=240.5 mg, t=10.5 h, and T_half=78 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 = 3.21 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 578.3008, "unit": "g"}

Answer: 578.3008

Prompt: A sample contains 3.21 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 3.21 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 1375 mL to 1.375 L.
3. Compute M = 2.05 / 1.375.
Final: {"answer": 1.4909, "unit": "mol/L"}

Answer: 1.4909

Prompt: A solution contains 2.05 mol of solute in 1375 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1375 mL to 1.375 L.
Compute M = 2.05 / 1.375.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(3.4)=34.14 and f(8.3)=56.68.
3. The interval width is 4.9.
Final: {"answer": 222.509, "unit": "area_units"}

Answer: 222.509

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=4.6x+18.5 from x=3.4 to x=8.3. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(3.4)=34.14 and f(8.3)=56.68.
The interval width is 4.9.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.22 and add it to the current value.
Final: {"answer": [-0.827, -1.509, -0.784], "unit": "temperature_units"}

Answer: -0.827

Answer: -1.509

Answer: -0.784

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2.8, right=-0.95; current interior values are [-1.85, -1.85, -0.3]. Use r=0.22. Return the next interior vector.

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


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=292 degrees and keep the unit as newtons.
Final: {"answer": [51.3211, -127.0242], "unit": "N"}

Answer: 51.3211

Answer: -127.0242

Prompt: A vector has magnitude 137 N at 292 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=292 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": [-9.5, 23.75], "unit": "dimensionless"}

Answer: -9.5

Answer: 23.75

Prompt: A line passes through points (4.75, -21.375) and (17.75, -144.875). Return [slope, intercept] for y = slope*x + intercept.

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


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

```python
def center_values_000322(values):
    if not values:
        raise ValueError('center_values requires at least one value')
    mean = sum(values) / len(values)
    return [value - mean for value in values]
```
Final: {"answer": "python_code", "function": "center_values_000322"}

Answer: python_code

Prompt: Write a Python function center_values_000322(values) that subtracts the arithmetic mean from each value and returns a new list. Raise ValueError for an empty input. Return a single fenced Python code block.

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

try:
    center_values_000322([])
    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)=23.5 and h=0.7, 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. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 4.3^2 and multiply by 0.5 * 21.9.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 202.4655, "unit": "J"}

Answer: 202.4655

Prompt: A body of mass 21.9 kg moves at 4.3 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 4.3^2 and multiply by 0.5 * 21.9.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=80 degrees through the sine of 2 theta.
3. Substitute v=49.6 m/s and g=9.81 m/s^2.
Final: {"answer": 85.7721, "unit": "m"}

Answer: 85.7721

Prompt: A projectile is launched on level ground with no air resistance at speed 49.6 m/s and angle 80 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=80 degrees through the sine of 2 theta.
Substitute v=49.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 0.52 * 8.314 * 379 / 118.1.
Final: {"answer": 13.874, "unit": "kPa"}

Answer: 13.874

Prompt: An ideal gas sample has n=0.52 mol, T=379 K, and V=118.1 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 0.52 * 8.314 * 379 / 118.1.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1700 * 1.245 * 63.
Final: {"answer": 133339.5, "unit": "J"}

Answer: 133339.5

Prompt: A 1700 g sample has specific heat 1.245 J/(g*K). How much heat is needed to raise its temperature by 63 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1700 * 1.245 * 63.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=294.5 mg, t=52 h, and T_half=16 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 30.9555, "unit": "mg"}

Answer: 30.9555

Prompt: A radioactive sample starts with 294.5 mg. Its half-life is 16 hours. How many mg remain after 52 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=294.5 mg, t=52 h, and T_half=16 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 = 5.69 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 102.5054, "unit": "g"}

Answer: 102.5054

Prompt: A sample contains 5.69 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 = 5.69 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 3410 mL to 3.41 L.
3. Compute M = 3.07 / 3.41.
Final: {"answer": 0.9003, "unit": "mol/L"}

Answer: 0.9003

Prompt: A solution contains 3.07 mol of solute in 3410 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3410 mL to 3.41 L.
Compute M = 3.07 / 3.41.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.81 y, so each step multiplies y by 1 - h*0.81.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 20.7251, "unit": "state_units"}

Answer: 20.7251

Prompt: Use explicit Euler on dy/dt = -0.81 y with y(0)=66.5, step size h=0.15, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.81 y, so each step multiplies y by 1 - h*0.81.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.12 and add it to the current value.
Final: {"answer": [0.648, 2.034, 0.53], "unit": "temperature_units"}

Answer: 0.648

Answer: 2.034

Answer: 0.53

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=0, right=-0.35; current interior values are [0.45, 2.55, 0.35]. Use r=0.12. Return the next interior vector.

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


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=125 degrees and keep the unit as newtons.
Final: {"answer": [-138.8055, 198.2348], "unit": "N"}

Answer: -138.8055

Answer: 198.2348

Prompt: A vector has magnitude 242 N at 125 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=125 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": [-3.4, -29.75], "unit": "dimensionless"}

Answer: -3.4

Answer: -29.75

Prompt: A line passes through points (-16.5, 26.35) and (-9.5, 2.55). 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. coulomb per second is ampere
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In student derivation case 938328, check dimensional consistency for I = Q / t. The left side has dimensions A and the right side has dimensions A. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
coulomb per second is ampere
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
def normalize_000340(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_000340"}

Answer: python_code

Prompt: Write a Python function normalize_000340(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_000340([2, 3, 5]); assert len(out) == 3

try:
    normalize_000340([1, -1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


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

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=11.5 and h=1.65, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 17.5 kg and a = 3.8 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 66.5, "unit": "N"}

Answer: 66.5

Prompt: A cart has mass 17.5 kg and measured acceleration 3.8 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 17.5 kg and a = 3.8 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 43.25^2 and multiply by 0.5 * 7.5.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 7014.6094, "unit": "J"}

Answer: 7014.6094

Prompt: A body of mass 7.5 kg moves at 43.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 43.25^2 and multiply by 0.5 * 7.5.
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=56 degrees through the sine of 2 theta.
3. Substitute v=52.5 m/s and g=9.81 m/s^2.
Final: {"answer": 260.5046, "unit": "m"}

Answer: 260.5046

Prompt: A projectile is launched on level ground with no air resistance at speed 52.5 m/s and angle 56 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=56 degrees through the sine of 2 theta.
Substitute v=52.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.23 * 8.314 * 602 / 50.7.
Final: {"answer": 318.8608, "unit": "kPa"}

Answer: 318.8608

Prompt: An ideal gas sample has n=3.23 mol, T=602 K, and V=50.7 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 3.23 * 8.314 * 602 / 50.7.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1495 * 3.77 * 90.
Final: {"answer": 507253.5, "unit": "J"}

Answer: 507253.5

Prompt: A 1495 g sample has specific heat 3.77 J/(g*K). How much heat is needed to raise its temperature by 90 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1495 * 3.77 * 90.


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=80.5 V and R=1.2 ohm.
Final: {"answer": 5400.2083, "unit": "W"}

Answer: 5400.2083

Prompt: A resistor has resistance 1.2 ohm and voltage 80.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=80.5 V and R=1.2 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=470 mg, t=181 h, and T_half=26 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 3.7711, "unit": "mg"}

Answer: 3.7711

Prompt: A radioactive sample starts with 470 mg. Its half-life is 26 hours. How many mg remain after 181 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=470 mg, t=181 h, and T_half=26 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.94 mol * 58.44 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 54.9336, "unit": "g"}

Answer: 54.9336

Prompt: A sample contains 0.94 mol of sodium chloride. Using molar mass 58.44 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.94 mol * 58.44 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.89 y, so each step multiplies y by 1 - h*0.89.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": 0.2577, "unit": "state_units"}

Answer: 0.2577

Prompt: Use explicit Euler on dy/dt = -0.89 y with y(0)=54, step size h=0.6, for 7 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.89 y, so each step multiplies y by 1 - h*0.89.
Apply that update for 7 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(8.3)=-57.81 and f(10.4)=-69.78.
3. The interval width is 2.1.
Final: {"answer": -133.9695, "unit": "area_units"}

Answer: -133.9695

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-5.7x+-10.5 from x=8.3 to x=10.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(8.3)=-57.81 and f(10.4)=-69.78.
The interval width is 2.1.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.28 and add it to the current value.
Final: {"answer": [0.108, -0.118, -0.232], "unit": "temperature_units"}

Answer: 0.108

Answer: -0.118

Answer: -0.232

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2, right=-1.1; current interior values are [-1.95, 1.45, -0.75]. 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=147 degrees and keep the unit as newtons.
Final: {"answer": [-288.5027, 187.3558], "unit": "N"}

Answer: -288.5027

Answer: 187.3558

Prompt: A vector has magnitude 344 N at 147 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=147 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": [-0.2, 21.0], "unit": "dimensionless"}

Answer: -0.2

Answer: 21.0

Prompt: A line passes through points (8.25, 19.35) and (9, 19.2). 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 406659, check dimensional consistency for C = n * V. The left side has dimensions mol/L and the right side has dimensions mol*L. Return consistent or inconsistent with a brief reason.

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

Answer: inconsistent


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

```python
def running_total_000358(values):
    total = 0
    out = []
    for value in values:
        total += value
        out.append(total)
    return out
```
Final: {"answer": "python_code", "function": "running_total_000358"}

Answer: python_code

Prompt: Write a Python function running_total_000358(values) that returns cumulative sums as a new list. It must not mutate the input list. Return a single fenced Python code block.

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


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: 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)=44 and h=0.45, 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 = 8.25 kg and a = 13.45 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 110.9625, "unit": "N"}

Answer: 110.9625

Prompt: A cart has mass 8.25 kg and measured acceleration 13.45 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 8.25 kg and a = 13.45 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=36 degrees through the sine of 2 theta.
3. Substitute v=30.1 m/s and g=9.81 m/s^2.
Final: {"answer": 87.8355, "unit": "m"}

Answer: 87.8355

Prompt: A projectile is launched on level ground with no air resistance at speed 30.1 m/s and angle 36 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=36 degrees through the sine of 2 theta.
Substitute v=30.1 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.01 * 8.314 * 242 / 119.0.
Final: {"answer": 84.7064, "unit": "kPa"}

Answer: 84.7064

Prompt: An ideal gas sample has n=5.01 mol, T=242 K, and V=119.0 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 5.01 * 8.314 * 242 / 119.0.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=206.5 V and R=163.4 ohm.
Final: {"answer": 260.9685, "unit": "W"}

Answer: 260.9685

Prompt: A resistor has resistance 163.4 ohm and voltage 206.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=206.5 V and R=163.4 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=326.5 mg, t=35.5 h, and T_half=77 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 237.1901, "unit": "mg"}

Answer: 237.1901

Prompt: A radioactive sample starts with 326.5 mg. Its half-life is 77 hours. How many mg remain after 35.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=326.5 mg, t=35.5 h, and T_half=77 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 = 2.57 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 46.2985, "unit": "g"}

Answer: 46.2985

Prompt: A sample contains 2.57 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 2.57 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2455 mL to 2.455 L.
3. Compute M = 3.795 / 2.455.
Final: {"answer": 1.5458, "unit": "mol/L"}

Answer: 1.5458

Prompt: A solution contains 3.795 mol of solute in 2455 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2455 mL to 2.455 L.
Compute M = 3.795 / 2.455.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.64 y, so each step multiplies y by 1 - h*0.64.
3. Apply that update for 11 steps and round the final state.
Final: {"answer": 28.2612, "unit": "state_units"}

Answer: 28.2612

Prompt: Use explicit Euler on dy/dt = -0.64 y with y(0)=58.5, step size h=0.1, 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) = -0.64 y, so each step multiplies y by 1 - h*0.64.
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(5.7)=19.81 and f(10.3)=44.19.
3. The interval width is 4.6.
Final: {"answer": 147.2, "unit": "area_units"}

Answer: 147.2

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=5.3x+-10.4 from x=5.7 to x=10.3. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(5.7)=19.81 and f(10.3)=44.19.
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=0.38 and add it to the current value.
Final: {"answer": [1.387, 1.911, 1.344], "unit": "temperature_units"}

Answer: 1.387

Answer: 1.911

Answer: 1.344

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=1.8; current interior values are [2.85, 0.6, 1.8]. 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. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 212.5 + 273.15.
Final: {"answer": 485.65, "unit": "K"}

Answer: 485.65

Prompt: Convert 212.5 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 212.5 + 273.15.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-3.7, 28.25], "unit": "dimensionless"}

Answer: -3.7

Answer: 28.25

Prompt: A line passes through points (-1, 31.95) and (8.5, -3.2). 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 safe_mean_000376(values):
    if not values:
        raise ValueError('safe_mean requires at least one value')
    return sum(values) / len(values)
```
Final: {"answer": "python_code", "function": "safe_mean_000376"}

Answer: python_code

Prompt: Write a Python function safe_mean_000376(values) that returns the arithmetic mean. It must raise ValueError for an empty input and must not mutate the input list. Return a single fenced Python code block.

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

try:
    safe_mean_000376([])
    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 routing scoring problem with 4091 items and budget 73190, 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 = 14 kg and a = 2.15 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 30.1, "unit": "N"}

Answer: 30.1

Prompt: A cart has mass 14 kg and measured acceleration 2.15 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 14 kg and a = 2.15 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 43.5^2 and multiply by 0.5 * 18.8.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 17787.15, "unit": "J"}

Answer: 17787.15

Prompt: A body of mass 18.8 kg moves at 43.5 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 43.5^2 and multiply by 0.5 * 18.8.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=71 degrees through the sine of 2 theta.
3. Substitute v=80.5 m/s and g=9.81 m/s^2.
Final: {"answer": 406.6912, "unit": "m"}

Answer: 406.6912

Prompt: A projectile is launched on level ground with no air resistance at speed 80.5 m/s and angle 71 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=71 degrees through the sine of 2 theta.
Substitute v=80.5 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 = 105 * 3.47 * 79.
Final: {"answer": 28783.65, "unit": "J"}

Answer: 28783.65

Prompt: A 105 g sample has specific heat 3.47 J/(g*K). How much heat is needed to raise its temperature by 79 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 105 * 3.47 * 79.


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=119.6 V and R=62.8 ohm.
Final: {"answer": 227.7732, "unit": "W"}

Answer: 227.7732

Prompt: A resistor has resistance 62.8 ohm and voltage 119.6 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=119.6 V and R=62.8 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=938.5 mg, t=276 h, and T_half=95.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 126.6024, "unit": "mg"}

Answer: 126.6024

Prompt: A radioactive sample starts with 938.5 mg. Its half-life is 95.5 hours. How many mg remain after 276 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=938.5 mg, t=276 h, and T_half=95.5 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 = 3.61 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 361.3141, "unit": "g"}

Answer: 361.3141

Prompt: A sample contains 3.61 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 = 3.61 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 4865 mL to 4.865 L.
3. Compute M = 1.74 / 4.865.
Final: {"answer": 0.3577, "unit": "mol/L"}

Answer: 0.3577

Prompt: A solution contains 1.74 mol of solute in 4865 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4865 mL to 4.865 L.
Compute M = 1.74 / 4.865.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.74 y, so each step multiplies y by 1 - h*0.74.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": 17.2919, "unit": "state_units"}

Answer: 17.2919

Prompt: Use explicit Euler on dy/dt = -0.74 y with y(0)=42.5, step size h=0.35, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.74 y, so each step multiplies y by 1 - h*0.74.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.1)=35.44 and f(3.3)=-4.52.
3. The interval width is 7.4.
Final: {"answer": 114.404, "unit": "area_units"}

Answer: 114.404

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-5.4x+13.3 from x=-4.1 to x=3.3. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.1)=35.44 and f(3.3)=-4.52.
The interval width is 7.4.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 1136.0 / 100.
Final: {"answer": 11.36, "unit": "m"}

Answer: 11.36

Prompt: Convert 1136 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 1136.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=278 degrees and keep the unit as newtons.
Final: {"answer": [52.0507, -370.3603], "unit": "N"}

Answer: 52.0507

Answer: -370.3603

Prompt: A vector has magnitude 374 N at 278 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=278 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-9.85, 22.75], "unit": "dimensionless"}

Answer: -9.85

Answer: 22.75

Prompt: A line passes through points (-10.5, 126.175) and (0.5, 17.825). 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. coulomb per second is ampere
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In sensor calibration case 858087, check dimensional consistency for I = Q / t. The left side has dimensions A and the right side has dimensions A. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
coulomb per second is ampere
The equation is valid only if both sides have identical dimensions.

Answer: consistent


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

```python
def running_total_000394(values):
    total = 0
    out = []
    for value in values:
        total += value
        out.append(total)
    return out
```
Final: {"answer": "python_code", "function": "running_total_000394"}

Answer: python_code

Prompt: Write a Python function running_total_000394(values) that returns cumulative sums as a new list. It must not mutate the input list. Return a single fenced Python code block.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 10.05 kg and a = 2.05 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 20.6025, "unit": "N"}

Answer: 20.6025

Prompt: A cart has mass 10.05 kg and measured acceleration 2.05 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 = 10.05 kg and a = 2.05 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 = 13.35^2 and multiply by 0.5 * 27.6.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 2459.4705, "unit": "J"}

Answer: 2459.4705

Prompt: A body of mass 27.6 kg moves at 13.35 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 13.35^2 and multiply by 0.5 * 27.6.
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=13 degrees through the sine of 2 theta.
3. Substitute v=52.4 m/s and g=9.81 m/s^2.
Final: {"answer": 122.6974, "unit": "m"}

Answer: 122.6974

Prompt: A projectile is launched on level ground with no air resistance at speed 52.4 m/s and angle 13 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=13 degrees through the sine of 2 theta.
Substitute v=52.4 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1035 * 1.28 * 6.
Final: {"answer": 7948.8, "unit": "J"}

Answer: 7948.8

Prompt: A 1035 g sample has specific heat 1.28 J/(g*K). How much heat is needed to raise its temperature by 6 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1035 * 1.28 * 6.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=136.5 V and R=178.2 ohm.
Final: {"answer": 104.5581, "unit": "W"}

Answer: 104.5581

Prompt: A resistor has resistance 178.2 ohm and voltage 136.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=136.5 V and R=178.2 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 1.65 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 297.2574, "unit": "g"}

Answer: 297.2574

Prompt: A sample contains 1.65 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 = 1.65 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.03 y, so each step multiplies y by 1 - h*0.03.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 19.5668, "unit": "state_units"}

Answer: 19.5668

Prompt: Use explicit Euler on dy/dt = -0.03 y with y(0)=34, step size h=1.5, for 12 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.03 y, so each step multiplies y by 1 - h*0.03.
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(4.7)=32.78 and f(8.5)=45.7.
3. The interval width is 3.8.
Final: {"answer": 149.112, "unit": "area_units"}

Answer: 149.112

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=3.4x+16.8 from x=4.7 to x=8.5. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(4.7)=32.78 and f(8.5)=45.7.
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": [-0.82, -0.06, 1.044], "unit": "temperature_units"}

Answer: -0.82

Answer: -0.06

Answer: 1.044

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=1.8; current interior values are [-1.7, 0.9, 0.5]. 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=139 degrees and keep the unit as newtons.
Final: {"answer": [-101.1311, 87.9119], "unit": "N"}

Answer: -101.1311

Answer: 87.9119

Prompt: A vector has magnitude 134 N at 139 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=139 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": [-0.4, 6.25], "unit": "dimensionless"}

Answer: -0.4

Answer: 6.25

Prompt: A line passes through points (0.25, 6.15) and (14.25, 0.55). Return [slope, intercept] for y = slope*x + intercept.

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


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

```python
import math

def unit_vector_000412(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_000412"}

Answer: python_code

Prompt: Write a Python function unit_vector_000412(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_000412(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)=5.5 and h=0.8, 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 = 8.1^2 and multiply by 0.5 * 13.15.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 431.3858, "unit": "J"}

Answer: 431.3858

Prompt: A body of mass 13.15 kg moves at 8.1 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 8.1^2 and multiply by 0.5 * 13.15.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=16 degrees through the sine of 2 theta.
3. Substitute v=13.7 m/s and g=9.81 m/s^2.
Final: {"answer": 10.1387, "unit": "m"}

Answer: 10.1387

Prompt: A projectile is launched on level ground with no air resistance at speed 13.7 m/s and angle 16 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=16 degrees through the sine of 2 theta.
Substitute v=13.7 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 = 1.08 * 8.314 * 497 / 86.1.
Final: {"answer": 51.8307, "unit": "kPa"}

Answer: 51.8307

Prompt: An ideal gas sample has n=1.08 mol, T=497 K, and V=86.1 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 = 1.08 * 8.314 * 497 / 86.1.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 960 * 3.03 * 109.
Final: {"answer": 317059.2, "unit": "J"}

Answer: 317059.2

Prompt: A 960 g sample has specific heat 3.03 J/(g*K). How much heat is needed to raise its temperature by 109 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 960 * 3.03 * 109.


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=196.6 V and R=98.4 ohm.
Final: {"answer": 392.8004, "unit": "W"}

Answer: 392.8004

Prompt: A resistor has resistance 98.4 ohm and voltage 196.6 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=196.6 V and R=98.4 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 6.71 mol * 44.01 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 295.3071, "unit": "g"}

Answer: 295.3071

Prompt: A sample contains 6.71 mol of carbon dioxide. Using molar mass 44.01 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 6.71 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 2530 mL to 2.53 L.
3. Compute M = 4.085 / 2.53.
Final: {"answer": 1.6146, "unit": "mol/L"}

Answer: 1.6146

Prompt: A solution contains 4.085 mol of solute in 2530 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2530 mL to 2.53 L.
Compute M = 4.085 / 2.53.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.68 y, so each step multiplies y by 1 - h*0.68.
3. Apply that update for 4 steps and round the final state.
Final: {"answer": 7.4923, "unit": "state_units"}

Answer: 7.4923

Prompt: Use explicit Euler on dy/dt = -0.68 y with y(0)=61, step size h=0.6, 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) = -0.68 y, so each step multiplies y by 1 - h*0.68.
Apply that update for 4 steps and round the final state.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.18 and add it to the current value.
Final: {"answer": [0.25, 0.125, -0.294], "unit": "temperature_units"}

Answer: 0.25

Answer: 0.125

Answer: -0.294

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2.9, right=2.9; current interior values are [-0.65, 0.8, -1.5]. Use r=0.18. Return the next interior vector.

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


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 3391.0 / 100.
Final: {"answer": 33.91, "unit": "m"}

Answer: 33.91

Prompt: Convert 3391 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 3391.0 / 100.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=284 degrees and keep the unit as newtons.
Final: {"answer": [76.6892, -307.5837], "unit": "N"}

Answer: 76.6892

Answer: -307.5837

Prompt: A vector has magnitude 317 N at 284 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=284 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [7.6, 0.75], "unit": "dimensionless"}

Answer: 7.6

Answer: 0.75

Prompt: A line passes through points (8, 61.55) and (15.5, 118.55). 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 threshold_flags_000430(values, threshold):
    return [value >= threshold for value in values]
```
Final: {"answer": "python_code", "function": "threshold_flags_000430"}

Answer: python_code

Prompt: Write a Python function threshold_flags_000430(values, threshold) that returns a list of booleans showing whether each value is greater than or equal to threshold. Return a single fenced Python code block.

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

assert threshold_flags_000430([1, 2, 3], 2) == [False, True, True]

assert threshold_flags_000430([], 5) == []

assert threshold_flags_000430([-1, 0], 0) == [False, True]


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 18.15^2 and multiply by 0.5 * 7.75.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1276.5122, "unit": "J"}

Answer: 1276.5122

Prompt: A body of mass 7.75 kg moves at 18.15 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 18.15^2 and multiply by 0.5 * 7.75.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=10 degrees through the sine of 2 theta.
3. Substitute v=67.8 m/s and g=9.81 m/s^2.
Final: {"answer": 160.2662, "unit": "m"}

Answer: 160.2662

Prompt: A projectile is launched on level ground with no air resistance at speed 67.8 m/s and angle 10 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=10 degrees through the sine of 2 theta.
Substitute v=67.8 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 = 2.31 * 8.314 * 367 / 22.2.
Final: {"answer": 317.4937, "unit": "kPa"}

Answer: 317.4937

Prompt: An ideal gas sample has n=2.31 mol, T=367 K, and V=22.2 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 = 2.31 * 8.314 * 367 / 22.2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1205 * 4.735 * 45.
Final: {"answer": 256755.375, "unit": "J"}

Answer: 256755.375

Prompt: A 1205 g sample has specific heat 4.735 J/(g*K). How much heat is needed to raise its temperature by 45 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1205 * 4.735 * 45.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=172.5 V and R=106.2 ohm.
Final: {"answer": 280.1907, "unit": "W"}

Answer: 280.1907

Prompt: A resistor has resistance 106.2 ohm and voltage 172.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=172.5 V and R=106.2 ohm.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=793.5 mg, t=255.5 h, and T_half=52 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 26.3298, "unit": "mg"}

Answer: 26.3298

Prompt: A radioactive sample starts with 793.5 mg. Its half-life is 52 hours. How many mg remain after 255.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=793.5 mg, t=255.5 h, and T_half=52 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 = 4.07 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 73.3211, "unit": "g"}

Answer: 73.3211

Prompt: A sample contains 4.07 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 4.07 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.38 y, so each step multiplies y by 1 - h*0.38.
3. Apply that update for 8 steps and round the final state.
Final: {"answer": 0.0395, "unit": "state_units"}

Answer: 0.0395

Prompt: Use explicit Euler on dy/dt = -0.38 y with y(0)=3, step size h=1.1, for 8 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.38 y, so each step multiplies y by 1 - h*0.38.
Apply that update for 8 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(-1.3)=-15.53 and f(5.1)=-21.29.
3. The interval width is 6.4.
Final: {"answer": -117.824, "unit": "area_units"}

Answer: -117.824

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-0.9x+-16.7 from x=-1.3 to x=5.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-1.3)=-15.53 and f(5.1)=-21.29.
The interval width is 6.4.


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=0.22 and add it to the current value.
Final: {"answer": [1.515, 1.728, 0.496], "unit": "temperature_units"}

Answer: 1.515

Answer: 1.728

Answer: 0.496

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.15, right=-1.7; current interior values are [1.35, 2.3, 0.65]. Use r=0.22. Return the next interior vector.

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


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [7.55, 7.75], "unit": "dimensionless"}

Answer: 7.55

Answer: 7.75

Prompt: A line passes through points (-18, -128.15) and (-11.75, -80.9625). 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 model card equation case 50710, 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_000448(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_000448"}

Answer: python_code

Prompt: Write a Python function unit_vector_000448(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_000448(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: launch_angle_or_vertical_velocity.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["launch_angle_or_vertical_velocity"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute projectile range on level ground. The prompt gives initial speed 12 m/s but does not give launch angle or vertical velocity. If there is insufficient information, say so explicitly.

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


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 0.85 kg and a = 10.5 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 8.925, "unit": "N"}

Answer: 8.925

Prompt: A cart has mass 0.85 kg and measured acceleration 10.5 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 0.85 kg and a = 10.5 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 = 3.25^2 and multiply by 0.5 * 7.15.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 37.7609, "unit": "J"}

Answer: 37.7609

Prompt: A body of mass 7.15 kg moves at 3.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 3.25^2 and multiply by 0.5 * 7.15.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=21 degrees through the sine of 2 theta.
3. Substitute v=63.1 m/s and g=9.81 m/s^2.
Final: {"answer": 271.5818, "unit": "m"}

Answer: 271.5818

Prompt: A projectile is launched on level ground with no air resistance at speed 63.1 m/s and angle 21 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=21 degrees through the sine of 2 theta.
Substitute v=63.1 m/s and g=9.81 m/s^2.