The nuclear activity of a radioactive element | vedclass

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Question

(B) The activity $A$ of a radioactive sample at time $t$ is given by $A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$, where $A_0$ is the initia

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$10$

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(B) The activity $A$ of a radioactive sample at time $t$ is given by $A = A_0 \left(\frac{1}{2}ight)^{\frac{t}{T_{1/2}}}$, where $A_0$ is the initial activity and $T_{1/2}$ is the half-life.Given that $A = \frac{1}{8} A_0$ at $t = 30\, 	ext{years}$.Substituting the values: $\frac{1}{8} A_0 = A_0 \left(\frac{1}{2}ight)^{\frac{30}{T_{1/2}}}$.$\left(\frac{1}{2}ight)^3 = \left(\frac{1}{2}ight)^{\frac{30}{T_{1/2}}}$.Equating the exponents: $3 = \frac{30}{T_{1/2}}$.Therefore, $T_{1/2} = \frac{30}{3} = 10\, 	ext{years}$.

The nuclear activity of a radioactive element becomes $\left(\frac{1}{8}\right)^{\text{th}}$ of its initial value in $30\, \text{years}$. The half-life of the radioactive element is $....\, \text{years}$.

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