Mixing tank problems
Problem 6.704 · medium
A tank holds \( \displaystyle 1000 \) L of water with \( \displaystyle 100 \) kg of salt dissolved. Brine with 1/4 kg of salt per liter flows in at \( \displaystyle 4 \) 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 \( \displaystyle 30 \) minutes, and the long-run amount.
- Rate in − rate out: S′ = 4·1/4 − (4/1000)S. The volume stays 1000 L.Reviewed
- \[ \frac{d}{d t} \left(250 - 150 e^{- \frac{t}{250}}\right) = \frac{3 e^{- \frac{t}{250}}}{5} \]This S(t) satisfies the equation ...✓ Proved
- \[ \left. 250 - 150 e^{\frac{\left(-1\right) t}{250}} \right|_{\substack{ t=0 }} = 100 \]... and starts at the given amount.✓ Proved
- \[ \left. 250 - 150 e^{\frac{\left(-1\right) t}{250}} \right|_{\substack{ t=30 }} = 250 - \frac{150}{e^{\frac{3}{25}}} \]S(30).✓ Proved
- \[ \lim_{t \to \infty}\left(250 - 150 e^{- \frac{t}{250}}\right) = 250 \]In the long run the tank matches the incoming concentration.✓ Proved
Answer \( S(t) = 250 - 150 e^{- \frac{t}{250}},\ S(30) = 250 - \frac{150}{e^{\frac{3}{25}}} \approx 116.962,\ \lim S = 250 \)
✓ 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 — The differential equation setup is correct, and the solution satisfies the initial condition and the ODE. The numerical evaluation and limit are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The differential equation setup is correct, and the solution satisfies the initial condition and the ODE. The numerical evaluation and limit are correct.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The differential equation setup correctly models the mixing process with constant volume. The proposed solution satisfies the ODE and initial condition, and the specific values and limit are calculated correctly.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.