Related rates
Problem 3.177 · medium
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 10 m³/min. How fast is the water level rising when the water is 6 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
- \[ \left. \frac{90}{\pi h^{2}} \right|_{\substack{ h=6 }} = \frac{5}{2 \pi} \]dh/dt = (dV/dt)/(dV/dh) at h = 6.✓ Proved
Answer \( \frac{5}{2 \pi} \ \text{m/min} \approx 0.7958\ \text{m/min} \)
✓ 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, computes the derivative accurately, and substitutes the given values correctly.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies related rates using similar triangles to relate radius and height, computes the derivative accurately, and substitutes the given values correctly.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies related rates using similar triangles and the chain rule. The algebraic steps and final result are correct.gpt-oss:20b: pass 2026-09-26
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 2026-09-26 with SymPy 1.14.0.