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Exponential growth and decay

Problem 5.222 · medium

A radioactive substance has a half-life of 30 years. Starting with 50 g, how much remains after 56 years, and when will only \frac{50}{3} g remain?

y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
k = -log(2)/30.
\[ \frac{25 \cdot 2^{\frac{2}{15}}}{2} \]
y(56).✓ Proved
\[ \frac{30 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \]
Solve y₀e^(kt) = 50/3 for t: t = ln(M/y₀)/k.✓ Proved

Answer \( y(56) = \frac{25 \cdot 2^{\frac{2}{15}}}{2} \approx 13.71,\quad t = \frac{30 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \approx 47.55 \)

Lines: 2 proved, 2 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.

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	the model evaluated, and the time found by a numerical root-finder

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution contains a critical arithmetic error in the final answer for y(56). The correct value is 25 * 2^(-26/30) ≈ 8.37 g, but the stated answer and the 'proved' equation yield ≈ 13.71 g (which corresponds to 25 * 2^(2/15), missing the division by 2 or incorrect exponent handling). Additionally, the formula for t is incorrect; it should be t = ln(1/3)/k, not ln(3)/k (though the numerical value 47.55 is correct for the magnitude, the sign in the log argument relative to k needs careful handling, but the primary failure is the mass calculation).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution contains a critical arithmetic error in the final answer for y(56). The correct value is 25 * 2^(-26/30) ≈ 8.37 g, but the stated answer and the 'proved' equation yield ≈ 13.71 g (which corresponds to 25 * 2^(2/15), missing the division by 2 or incorrect exponent handling). Additionally, the formula for t is incorrect; it should be t = ln(1/3)/k, not ln(3)/k (though the numerical value 47.55 is correct for the magnitude, the sign in the log argument relative to k needs careful handling, but the primary failure is the mass calculation).
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the correct value for y(56). The stated answer approximates 13.71, but the correct value is approximately 16.42 g. The formula in line 3 is incorrect (it divides by 2 unnecessarily or uses the wrong exponent logic), leading to a wrong numerical result.
gpt-oss:20b: pass 2026-10-04

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/exponential_models, checked 2026-10-04 with SymPy 1.14.0.