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Mixing tank problems

Problem 6.786 · medium

A tank holds \( \displaystyle 500 \) L of water with \( \displaystyle 10 \) kg of salt dissolved. Brine with 1/4 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′ = 2·1/4 − (2/500)S. The volume stays 500 L.
Reviewed
\[ \frac{d}{d t} \left(125 - 115 e^{- \frac{t}{250}}\right) = \frac{23 e^{- \frac{t}{250}}}{50} \]
This S(t) satisfies the equation ...✓ Proved
\[ \left. 125 - 115 e^{\frac{\left(-1\right) t}{250}} \right|_{\substack{ t=0 }} = 10 \]
... and starts at the given amount.✓ Proved
\[ \left. 125 - 115 e^{\frac{\left(-1\right) t}{250}} \right|_{\substack{ t=10 }} = 125 - \frac{115}{e^{\frac{1}{25}}} \]
S(10).✓ Proved
\[ \lim_{t \to \infty}\left(125 - 115 e^{- \frac{t}{250}}\right) = 125 \]
In the long run the tank matches the incoming concentration.✓ Proved

Answer \( S(t) = 125 - 115 e^{- \frac{t}{250}},\ S(10) = 125 - \frac{115}{e^{\frac{1}{25}}} \approx 14.509,\ \lim S = 125 \)

Lines: 4 proved, 1 reviewed. 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	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 — The solution correctly sets up the differential equation, verifies the general 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 2026-11-02 — The solution correctly sets up the differential equation, verifies the general solution satisfies the ODE and initial condition, and correctly computes the specific value and limit.
gpt-oss:20b: pass 2026-11-02
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The stated answer for S(10) is approximately 14.509, but the correct value is approximately 12.41. The solution correctly derives the formula and the symbolic expression for S(10), but the final numerical approximation provided in the problem statement is incorrect, leading to a false result.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The stated answer for S(10) is approximately 14.509, but the correct value is approximately 12.51. The solution likely confused the exponent -1/25 with -1/2.5 or made a calculation error, resulting in a significantly incorrect numerical answer.
gpt-oss:20b: pass 2026-11-01

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 2026-11-02 with SymPy 1.14.0.