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Mixing tank problems

Problem 6.785 · medium

A tank holds \( \displaystyle 100 \) L of water with \( \displaystyle 10 \) 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 \( \displaystyle 20 \) minutes, and the long-run amount.

Rate in − rate out: S′ = 5·1/10 − (5/100)S. The volume stays 100 L.
\[ \frac{d}{d t} 10 = 0 \]
This S(t) satisfies the equation ...✓ Proved
\[ \left. 10 \right|_{\substack{ t=0 }} = 10 \]
... and starts at the given amount.✓ Proved
\[ \left. 10 \right|_{\substack{ t=20 }} = 10 \]
S(20).✓ Proved
\[ \lim_{t \to \infty} 10 = 10 \]
In the long run the tank matches the incoming concentration.✓ Proved

Answer \( S(t) = 10,\ S(20) = 10 \approx 10.000,\ \lim S = 10 \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The differential equation setup is incorrect: the rate out should be (5/100)S, but the check in line 2 implies a different equation or the solution S

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-11-02 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The differential equation setup is incorrect: the rate out should be (5/100)S, but the check in line 2 implies a different equation or the solution S
gpt-oss:20b: pass 2026-11-02
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution claims S(t) = 10 is the solution, but this is incorrect. The initial condition is S(0) = 10, but the derivative S' = 0.5 - 0.05S is not zero at S=10 (it is 0). Wait, S' = 0.5 - 0.05(10) = 0. So S(t)=10 is actually the equilibrium solution. Let's re-read carefully. S' = 5*(1/10) - (5/100)S = 0.5 - 0.05S. If S(t) = 10, S' = 0.5 - 0.5 = 0. So S(t)=10 is a constant solution. Does it satisfy the initial condition? S(0)=10. Yes. So S(t)=10 for all t. The stated answer is correct. Why did I think it was wrong? Ah, usually these problems have S(0) != equilibrium. Here S(0)=10 and equilibrium is 10. So the solution is trivial. The steps are correct. The sentence in step 1 sets up the ODE correctly. Step 2 checks the ODE. Step 3 checks IC. Step 4 and 5 check values. It seems correct. Wait, let me double check the problem statement. 100 L, 10 kg salt. Inflow 5 L/min, 1/10 kg/L. Outflow 5 L/min. Equilibrium concentration = inflow concentration = 1/10 kg/L. Equilibrium amount = 100 L * 1/10 kg/L = 10 kg. Initial amount = 10 kg. So the system starts at equilibrium. S(t) = 10 is correct. The solution is sound.
gpt-oss:20b: fail (misleading) 2026-11-01 — The solution incorrectly states the derivative of the constant solution as "-10*1/20 + (1/10)*5", which is a confusing and unnecessary expression. The correct check is simply that S′=0 and the RHS evaluates to 0 for S=10. The rest of the steps are fine.
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution fails to solve the differential equation. It incorrectly assumes the constant function S(t)=10 is the solution, but S(0)=10 is not the initial condition (S(0)=10 is given, but the derivative at t=0 is 5/2 - 5/10 = 1.5, not 0). The correct solution is S(t) = 10 - 9e^{-t/20}. The provided answer S(t)=10 is mathematically incorrect for the transient behavior.
gpt-oss:20b: fail (error) 2026-11-01 — Sentence 2 incorrectly applies the differential equation to a constant solution. The derivative of the proposed solution S(t)=10 is 0, but the equation requires S′=0.5−0.05S, which is not satisfied by S=10. Thus the solution is not correct.

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.