Chapter 3.1, Problem 8E

1. (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.
2. (b) Show that the solution of the initial-value problem in part (a) can be written A(t) = A₀2^−t/T.
3. (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 $\frac{1}{8}{A}_0$?