(a) Consider the initial-value problem dAdt=kA,A(0)=A0as the model for the decay of a radioactive substance. Show that, in general, the half-life Tof the substance isT=-ln(2)k.The solution ofdAdt=kA,A(0)=A0isA(t)=. Letting A=and solving for t,we obtain the half-life T=-ln(2)k.(b) Show that the solution of the initial-value problem in part (a) can be written A(t)=A02-tT.Since k=,we have A(t)=A0ekt=A02-tT.(c)If a radioactive substance has the half-life T given in part (a), how long will it take an initial amount A0of the substance to decay to18A0?An initial amount A0 will decay to18A0inhalf-lives.