Home›Calculus 2›Mixing tank problems›Problem 6.708

Mixing tank problems

Problem 6.708 · medium

A tank holds \( \displaystyle 500 \) L of water with \( \displaystyle 100 \) kg of salt dissolved. Brine with 1/10 kg of salt per liter flows in at \( \displaystyle 5 \) 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′ = 5·1/10 − (5/500)S. The volume stays 500 L.
Reviewed
\[ \frac{d}{d t} \left(50 + 50 e^{- \frac{t}{100}}\right) = - \frac{e^{- \frac{t}{100}}}{2} \]
This S(t) satisfies the equation ...✓ Proved
\[ \left. 50 e^{\frac{\left(-1\right) t}{100}} + 50 \right|_{\substack{ t=0 }} = 100 \]
... and starts at the given amount.✓ Proved
\[ \left. 50 e^{\frac{\left(-1\right) t}{100}} + 50 \right|_{\substack{ t=60 }} = \frac{50}{e^{\frac{3}{5}}} + 50 \]
S(60).✓ Proved
\[ \lim_{t \to \infty}\left(50 + 50 e^{- \frac{t}{100}}\right) = 50 \]
In the long run the tank matches the incoming concentration.✓ Proved

Answer \( S(t) = 50 + 50 e^{- \frac{t}{100}},\ S(60) = \frac{50}{e^{\frac{3}{5}}} + 50 \approx 77.441,\ \lim S = 50 \)

✓ 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 2026-10-19
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-18 — The setup correctly models the mixing problem with the right rate in, rate out, and initial conditions. The solution satisfies the differential equation and boundary conditions as verified by the algebraic checks.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-18
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.