Mixing tank problems
Problem 6.452 · medium
A tank holds \( \displaystyle 200 \) L of water with \( \displaystyle 0 \) kg of salt dissolved. Brine with 1/4 kg of salt per liter flows in at \( \displaystyle 20 \) L/min, and the well-mixed solution drains at the same rate. Find the salt \( \displaystyle S(t) \) in kg after \( \displaystyle t \) minutes, the amount after \( \displaystyle 60 \) minutes, and the long-run amount.
- Rate in − rate out: S′ = 20·1/4 − (20/200)S. The volume stays 200 L.Reviewed
- \[ \frac{d}{d t} \left(50 - 50 e^{- \frac{t}{10}}\right) = 5 e^{- \frac{t}{10}} \]This S(t) satisfies the equation ...✓ Proved
- \[ \left. 50 - 50 e^{\frac{\left(-1\right) t}{10}} \right|_{\substack{ t=0 }} = 0 \]... and starts at the given amount.✓ Proved
- \[ \left. 50 - 50 e^{\frac{\left(-1\right) t}{10}} \right|_{\substack{ t=60 }} = 50 - \frac{50}{e^{6}} \]S(60).✓ Proved
- \[ \lim_{t \to \infty}\left(50 - 50 e^{- \frac{t}{10}}\right) = 50 \]In the long run the tank matches the incoming concentration.✓ Proved
Answer \( S(t) = 50 - 50 e^{- \frac{t}{10}},\ S(60) = 50 - \frac{50}{e^{6}} \approx 49.876,\ \lim S = 50 \)
✓ Nihil obstat Lines: 4 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | mpmath's ODE solver integrates the tank from t = 0 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14
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/mixing_tank, checked 2026-10-17 with SymPy 1.14.0.