Related rates
Problem 3.100 · medium
Water is poured into an inverted conical tank, 10 m tall with top radius 4 m, at 10 m³/min. How fast is the water level rising when the water is 8 m deep?
- By similar triangles the water's radius is r = (4/10)h, so V = (1/3)π((4/10)h)²h.Reviewed
- \[ \frac{d}{d h} \frac{4 \pi h^{3}}{75} = \frac{4 \pi h^{2}}{25} \]dV/dt = (dV/dh)(dh/dt).✓ Proved
- \[ \left. \frac{125}{2 \pi h^{2}} \right|_{\substack{ h=8 }} = \frac{125}{128 \pi} \]dh/dt = (dV/dt)/(dV/dh) at h = 8.✓ Proved
Answer \( \frac{125}{128 \pi} \ \text{m/min} \approx 0.3108\ \text{m/min} \)
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. 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 models the geometry using similar triangles, derives the volume formula, and applies the chain rule to find the rate of change of height. The final calculation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly models the geometry using similar triangles, derives the volume formula, and applies the chain rule to find the rate of change of height. The final calculation is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies related rates using similar triangles to eliminate the radius variable, and the algebraic steps are sound.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to substitute the given rate dV/dt = 10 m³/min into the final calculation. It computes the factor 1/(dV/dh) but omits multiplying by dV/dt, resulting in an answer that is off by a factor of 10.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.