(a) Consider the initial-value problem dA/dt = kA, A(0) = A0 as the model for the de...

(a) Consider the initial-value problem dA/dt = kA, A(0) = A0 as the model for the decay of a radioactive substance. Show that, in general, the half-life T of the substance is T= -(ln 2)/k.

(b) Show that the solution of the initial-value problem in part (a) can be written A(t) = A₀2^-t/T.

(c) If a radioactive substance has the half-life T given in part (a), how long will it take an initial amount A0 of the substance to decay to