Differentials
Problem 3.436 · easy
The radius of a sphere changes from \( \displaystyle 8 \) to \( \displaystyle \frac{161}{20} \). 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
- \[ \frac{\left. 4 \pi x^{2} \right|_{\substack{ x=8 }}}{20} = \frac{64 \pi}{5} \]Substitute x and dx.✓ Proved
- The exact change is 77281*pi/6000 ≈ 40.4642; the differential is the tangent-line estimate of it.Reviewed
Answer \( \frac{64 \pi}{5} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the derivative of the volume formula and substitutes the given radius and change in radius (dx = 1/20) to find the differential estimate. The final sentence correctly contextualizes the result as an approximation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the derivative of the volume formula and substitutes the given radius and change in radius (dx = 1/20) to find the differential estimate. The final sentence correctly contextualizes the result as an approximation.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the derivative of the volume formula, substitutes the initial radius and the change in radius (dx = 0.05) to find dV, and correctly interprets the result as a linear approximation.
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 2026-10-06 with SymPy 1.14.0.