Differentials
Problem 3.385 · easy
The radius of a sphere changes from \( \displaystyle 3 \) to \( \displaystyle \frac{301}{100} \). 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
- \[ \frac{\left. 4 \pi x^{2} \right|_{\substack{ x=3 }}}{100} = \frac{9 \pi}{25} \]Substitute x and dx.✓ Proved
- The exact change is 270901*pi/750000 ≈ 1.13475; the differential is the tangent-line estimate of it.
Answer \( \frac{9 \pi}{25} \)
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 exact change is incorrectly computed; the correct value is (4π/3)(301/100)^3−(4π/3)(3)^3 = 270901π/750000? Actually the correct exact change is 270901π/750000? No, the correct exact change is 270901π/750000? Wait, the exact change is (4π/3)(301/100)^3−(4π/3)(3)^3 = 270901π/750000? This is incorrect; the correct exact change is 270901π/750000? The differential estimate 9π/25 is correct, but the sentence claiming the exact change is wrong.qwen3.6:27b-mlx: pass — The solution correctly applies the differential formula dV = 4πr²dr with r=3 and dr=1/100. The final sentence correctly identifies the differential as a tangent-line approximation.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the differential formula dV = 4πr²dr with r=3 and dr=1/100. The final sentence correctly identifies the differential as a tangent-line approximation.gpt-oss:20b: fail (misleading) 2026-10-04 — The exact change is incorrectly computed; the correct value is (4π/3)(301/100)^3−(4π/3)(3)^3 = 270901π/750000? Actually the correct exact change is 270901π/750000? No, the correct exact change is 270901π/750000? Wait, the exact change is (4π/3)(301/100)^3−(4π/3)(3)^3 = 270901π/750000? This is incorrect; the correct exact change is 270901π/750000? The differential estimate 9π/25 is correct, but the sentence claiming the exact change is wrong.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the differential formula dV = 4πr²dr with r=3 and dr=1/100. The final sentence accurately contextualizes the result as a linear approximation.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.