Question

A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be _______ hours.

Answer 1

Correct Answer: 15

The problem involves the concept of radioactive decay, characterized by its half-life, which is the time required for a quantity of a radioactive substance to reduce to half its initial amount. Given:

Initial intensity: 64 times the safe level.

Half-life \( T_{1/2} = 2.5 \) hours (2 hours 30 minutes).

Let \( I_0 \) be the initial intensity, and \( I_s \) the safe level. After time \( t \), the intensity is \( I(t) = I_0 \times (1/2)^{t/T_{1/2}} \).

We want \( I(t) = I_s \).

\[ \frac{I_0}{I_s} \times (1/2)^{t/T_{1/2}} = 1 \]

\[ 64 \times (1/2)^{t/2.5} = 1 \]

Solve for \( t \):

\[ (1/2)^{t/2.5} = 1/64 \]

\[ 2^{t/2.5} = 64 \]

\[ 2^{t/2.5} = 2^6 \]

Equating exponents,

\[ \frac{t}{2.5} = 6 \]

\[ t = 6 \times 2.5 = 15 \text{ hours} \]

Thus, the minimum time required is 15 hours. This result fits within the given range, confirming its correctness.