Related rates
Problem 3.221 · medium
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 5 m³/min. How fast is the water level rising when the water is 3 m deep?
- By similar triangles the water's radius is r = (3/9)h, so V = (1/3)π((3/9)h)²h.Reviewed
- \[ \frac{d}{d h} \frac{\pi h^{3}}{27} = \frac{\pi h^{2}}{9} \]dV/dt = (dV/dh)(dh/dt).✓ Proved
- \[ \left. \frac{45}{\pi h^{2}} \right|_{\substack{ h=3 }} = \frac{5}{\pi} \]dh/dt = (dV/dt)/(dV/dh) at h = 3.✓ Proved
Answer \( \frac{5}{\pi} \ \text{m/min} \approx 1.592\ \text{m/min} \)
✓ 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the situation was stepped forward and back by a microsecond and the quantity differenced numerically |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies related rates using similar triangles to relate radius and height, derives the volume formula, and computes the derivative and final value accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies related rates using similar triangles to relate radius and height, derives the volume formula, and computes the derivative and final value accurately.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies related rates using similar triangles to relate radius and height, derives the volume formula, differentiates with respect to time, and substitutes the correct values to find the rate of change of the water level.gpt-oss:20b: pass 2026-09-27
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/related_rates, checked 2026-09-27 with SymPy 1.14.0.