In a radioactive material, fraction of active material remaining after time t is 9/16. The fraction that was remaining after t/2 is

Solution

Physics sir · 3 steps

Decay is exponential, so the fraction left at half the time is the square root of the fraction at t.

1. Decay law. The fraction of active material left after time t is N/N₀ = e^(-λ t). Here e^(-λ t) = 9/16.
2. Halve the time. After t/2 the exponent is halved. Halving an exponent means taking the square root of the whole quantity.

   N/N₀|_(t/2) = e^(-λ t/2) = (e^(-λ t))^(1/2) = √(9/16) = 3/4

3. So. The fraction remaining after t/2 is 3/4, option (a).

The idea to keep. In exponential decay the fraction left multiplies over equal time steps. Twice the time squares the fraction, and half the time takes its square root. It never scales linearly.

Answer from the NTA final answer key. Solution by PrepWiser.