A nuclear power plant supplying electrical power to a village uses a radioactive material of half life $T$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is $12.5$ of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $n T$ years, then the value of $n$ is

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Solution

Initially the radioactivity of the plant is $A_{0}$ and power requirment of the town is $A .$
So the power devloped by the plant to match the town power;
$\frac{A_{0}}{8} = A_{0} e^{- λ t} \Rightarrow \frac{A_{0}}{2^{t / T}}$
$\frac{A_{0}}{8} = \frac{A_{0}}{2^{t / T}}$
$8 = 2^{t / T} 2^{3} = 2^{t / T}$
So, $\Rightarrow 3 = \frac{t}{T} t = 3 T$

Hence the value of $n = 3.$

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Initially the radioactivity of the plant is A0 and power requirment of the town is A. So the power devloped by the plant to match the town power; A08=A0e−λt⇒A02t/T A08=A02t/T 8=2t/T23=2t/T So, ⇒3=tTt=3T Hence the value of n=3.