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Mixing tank problems

Problem 6.445 · medium

A tank holds \( \displaystyle 200 \) L of water with \( \displaystyle 50 \) kg of salt dissolved. Brine with 1/10 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 \( \displaystyle 20 \) minutes, and the long-run amount.

Rate in − rate out: S′ = 4·1/10 − (4/200)S. The volume stays 200 L.
Reviewed
\[ \frac{d}{d t} \left(20 + 30 e^{- \frac{t}{50}}\right) = - \frac{3 e^{- \frac{t}{50}}}{5} \]
This S(t) satisfies the equation ...✓ Proved
\[ \left. 30 e^{\frac{\left(-1\right) t}{50}} + 20 \right|_{\substack{ t=0 }} = 50 \]
... and starts at the given amount.✓ Proved
\[ \left. 30 e^{\frac{\left(-1\right) t}{50}} + 20 \right|_{\substack{ t=20 }} = 20 + \frac{30}{e^{\frac{2}{5}}} \]
S(20).✓ Proved
\[ \lim_{t \to \infty}\left(20 + 30 e^{- \frac{t}{50}}\right) = 20 \]
In the long run the tank matches the incoming concentration.✓ Proved

Answer \( S(t) = 20 + 30 e^{- \frac{t}{50}},\ S(20) = 20 + \frac{30}{e^{\frac{2}{5}}} \approx 40.110,\ \lim S = 20 \)

✓ 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-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
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly sets up the differential equation, 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.