Mixing tank problems
Problem 6.455 · medium
A tank holds \( \displaystyle 500 \) L of water with \( \displaystyle 10 \) kg of salt dissolved. Pure water flows in at \( \displaystyle 10 \) 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′ = 10·0 − (10/500)S. The volume stays 500 L.Reviewed
- \[ \frac{d}{d t} 10 e^{- \frac{t}{50}} = - \frac{e^{- \frac{t}{50}}}{5} \]This S(t) satisfies the equation ...✓ Proved
- \[ \left. 10 e^{\frac{\left(-1\right) t}{50}} \right|_{\substack{ t=0 }} = 10 \]... and starts at the given amount.✓ Proved
- \[ \left. 10 e^{\frac{\left(-1\right) t}{50}} \right|_{\substack{ t=60 }} = \frac{10}{e^{\frac{6}{5}}} \]S(60).✓ Proved
- \[ \lim_{t \to \infty}\left(10 e^{- \frac{t}{50}}\right) = 0 \]In the long run the tank matches the incoming concentration.✓ Proved
Answer \( S(t) = 10 e^{- \frac{t}{50}},\ S(60) = \frac{10}{e^{\frac{6}{5}}} \approx 3.012,\ \lim S = 0 \)
✓ 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 the mixing problem, 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 2026-10-17 — 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 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14
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.