Question

(X) A radioactive material decays at a rate proportional to the population present at time t. After 6 hours, the material has decreased by 87.5% (remained 12.5%). What is the half-life of this material? 24 a) d) 2 c) b) 4

Answer 1

Given: A radioactive material decays at a rate proportional to the population(amount) present at time t. Let A(t) be the amount at time t. then dA(t)/dt = -KA where k is constant of proportionality rightarrow dA(t)/A = -kdt integrating, we get ln A = -kt + c rightarrow A = e^-kt + c rightarrow A = e^-kt.e6c rightarrow A = A_0 e^-kt ------(1) where A_0 = e6c is the initial amount of the material given, after 6 hours, the material has decreased by 87.5% (remained 12.5%) i.e A = 12.5% of A_0 = 0.125 A_0 at t = 6 putting these values in 1 0.125 A_0 = A_0 e^- 6k 0.125 = e^-6k taking log(natural) both sides l(0.125) = -6k rightarrow - 2.-7944 = -6k rightarrow k = 0.3465736 putting this value of k in 1 A = A_0 e^- 0.3465736 t putting A = 1/2 A_0