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(C) The power generated by the nuclear plant is directly proportional to the activity $A$ of the radioactive fuel. Let $P_0$ be the initial power and

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$3$

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(C) The power generated by the nuclear plant is directly proportional to the activity $A$ of the radioactive fuel. Let $P_0$ be the initial power and $P(t)$ be the power at time $t$.The activity at time $t$ is given by $A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$,where $T$ is the half-life.Given that the village's power requirement $P_{req}$ is $12.5 \%$ of the initial power $P_0$,we have $P_{req} = 0.125 P_0 = \frac{1}{8} P_0$.The plant can meet the needs as long as $P(t) \ge P_{req}$. The maximum time $t = nT$ occurs when $P(nT) = P_{req}$.Thus,$P_0 \left(\frac{1}{2}\right)^{\frac{nT}{T}} = \frac{1}{8} P_0$.$\left(\frac{1}{2}\right)^n = \frac{1}{8} = \left(\frac{1}{2}\right)^3$.Comparing the exponents,we get $n = 3$.

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