Mixing tank problems
Problem 6.587 · medium
A tank holds \( \displaystyle 500 \) L of water with \( \displaystyle 20 \) kg of salt dissolved. Brine with 1/2 kg of salt per liter flows in at \( \displaystyle 2 \) 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 \( \displaystyle 10 \) minutes, and the long-run amount.
- Rate in − rate out: S′ = 2·1/2 − (2/500)S. The volume stays 500 L.
- \[ \frac{d}{d t} \left(250 - 230 e^{- \frac{t}{250}}\right) = \frac{23 e^{- \frac{t}{250}}}{25} \]This S(t) satisfies the equation ...✓ Proved
- \[ \left. 250 - 230 e^{\frac{\left(-1\right) t}{250}} \right|_{\substack{ t=0 }} = 20 \]... and starts at the given amount.✓ Proved
- \[ \left. 250 - 230 e^{\frac{\left(-1\right) t}{250}} \right|_{\substack{ t=10 }} = 250 - \frac{230}{e^{\frac{1}{25}}} \]S(10).✓ Proved
- \[ \lim_{t \to \infty}\left(250 - 230 e^{- \frac{t}{250}}\right) = 250 \]In the long run the tank matches the incoming concentration.✓ Proved
Answer \( S(t) = 250 - 230 e^{- \frac{t}{250}},\ S(10) = 250 - \frac{230}{e^{\frac{1}{25}}} \approx 29.018,\ \lim S = 250 \)
Lines: 4 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The stated answer for S(10) is approximately 29.018, but the correct value is approximately 220.98. The solution likely confused the result with the amount of salt removed or made a sign error in the final arithmetic, as S(10) must be between 20 and 250.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer for S(10) is approximately 29.018, but the correct value is approximately 220.98. The solution likely confused the result with the amount of salt removed or made a sign error in the final arithmetic, as S(10) must be between 20 and 250.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer for S(10) is approximately 29.018, but the correct value is approximately 229.02. The solution likely confused the transient term (230) with the total amount or made a decimal error, resulting in a value far below the initial amount of 20 kg, which is impossible since salt is being added.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer for S(10) is approximately 29.018, but the correct value is approximately 220.98. The solution likely confused the transient term (230) with the result, or calculated 250 - 220.98 instead of 250 - 29.018. The formula S(t) is correct, but the numerical evaluation in the stated answer is wrong.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.