Differentials
Problem 3.493 · medium
The radius of a sphere changes from \( \displaystyle 12 \) to \( \displaystyle \frac{119}{10} \). Use differentials to estimate the change in its volume, \( \displaystyle dV \).
- \[ \frac{d}{d x} \frac{4 \pi x^{3}}{3} = 4 \pi x^{2} \]dy = f′(x) dx.✓ Proved
- \[ - \frac{\left. 4 \pi x^{2} \right|_{\substack{ x=12 }}}{10} = - \frac{288 \pi}{5} \]Substitute x and dx.✓ Proved
- The exact change is -42841*pi/750 ≈ -179.452; the differential is the tangent-line estimate of it.
Answer \( - \frac{288 \pi}{5} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | central difference quotient of f at x times dx |
Reviewers
gpt-oss:20b: fail (misleading) — The sentence gives the exact change as -42841π/750, but the correct exact change is -42801π/750. The differential estimate is correct, but the stated exact value is false.qwen3.6:27b-mlx: pass — The solution correctly applies the differential formula dV = 4πx²dx with x=12 and dx=-1/10. The final sentence accurately contextualizes the result as an estimate.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-07 — The sentence gives the exact change as -42841π/750, but the correct exact change is -42801π/750. The differential estimate is correct, but the stated exact value is false.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the differential formula dV = 4πx²dx with x=12 and dx=-1/10. The final sentence accurately contextualizes the result as an estimate.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the derivative of the volume formula, substitutes the given radius and change in radius (dx = -1/10), and provides the correct estimated change in volume. The additional context about the exact change is accurate and helpful.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/differentials, checked 2026-10-07 with SymPy 1.14.0.