Mixing tank problems
Problem 6.706 · medium
A tank holds \( \displaystyle 500 \) L of water with \( \displaystyle 50 \) kg of salt dissolved. Pure water 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′ = 20·0 − (20/500)S. The volume stays 500 L.Reviewed
- \[ \frac{d}{d t} 50 e^{- \frac{t}{25}} = - 2 e^{- \frac{t}{25}} \]This S(t) satisfies the equation ...✓ Proved
- \[ \left. 50 e^{\frac{\left(-1\right) t}{25}} \right|_{\substack{ t=0 }} = 50 \]... and starts at the given amount.✓ Proved
- \[ \left. 50 e^{\frac{\left(-1\right) t}{25}} \right|_{\substack{ t=60 }} = \frac{50}{e^{\frac{12}{5}}} \]S(60).✓ Proved
- \[ \lim_{t \to \infty}\left(50 e^{- \frac{t}{25}}\right) = 0 \]In the long run the tank matches the incoming concentration.✓ Proved
Answer \( S(t) = 50 e^{- \frac{t}{25}},\ S(60) = \frac{50}{e^{\frac{12}{5}}} \approx 4.536,\ \lim S = 0 \)
✓ 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 solution correctly sets up the differential equation for a mixing problem with constant volume, verifies the solution satisfies the ODE and initial condition, and correctly computes the specific value and limit.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up the differential equation for a mixing problem with constant volume, verifies the solution satisfies the ODE and initial condition, and correctly computes the specific value and limit.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly sets up the differential equation for a mixing problem with constant volume, verifies the solution satisfies the ODE and initial condition, and correctly computes the specific value and limit.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly sets up the differential equation for the mixing problem, verifies the solution satisfies the ODE and initial condition, and correctly computes the specific value and limit.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.