Related rates
Problem 3.341 · medium
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 3 m³/min. How fast is the water level rising when the water is 7 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
- \[ \left. \frac{27}{\pi h^{2}} \right|_{\substack{ h=7 }} = \frac{27}{49 \pi} \]dh/dt = (dV/dt)/(dV/dh) at h = 7.✓ Proved
Answer \( \frac{27}{49 \pi} \ \text{m/min} \approx 0.1754\ \text{m/min} \)
Lines: 2 proved, 1 reviewed. 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 | 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 correctly computes the derivative and final value.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies related rates using similar triangles to relate radius and height, derives the volume formula, and correctly computes the derivative and final value.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — The solution omits the statement that the water is poured at 3 m³/min; without this the computation of dh/dt is incomplete and could mislead a student.
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 2026-10-03 with SymPy 1.14.0.