Related rates
Problem 3.15 · easy
Water is poured into an inverted conical tank, 6 m tall with top radius 2 m, at 5 m³/min. How fast is the water level rising when the water is 4 m deep?
- By similar triangles the water's radius is r = (2/6)h, so V = (1/3)π((2/6)h)²h.Reviewed
- \[ \frac{d}{d h} \frac{\pi h^{3}}{27} = \frac{\pi h^{2}}{9} \]dV/dt = (dV/dh)(dh/dt).✓ Proved
- \[ \left. \frac{45}{\pi h^{2}} \right|_{\substack{ h=4 }} = \frac{45}{16 \pi} \]dh/dt = (dV/dt)/(dV/dh) at h = 4.✓ Proved
Answer \( \frac{45}{16 \pi} \ \text{m/min} \approx 0.8952\ \text{m/min} \)
✓ 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. How we verify
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 and the chain rule. The algebraic steps are verified, and the final answer matches the stated result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies related rates using similar triangles and the chain rule. The algebraic steps are verified, and the final answer matches the stated result.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies related rates using similar triangles and the chain rule. The algebraic steps and final substitution are accurate.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-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 2026-09-26
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 2026-09-28 with SymPy 1.14.0.