Chapter 3.1, Problem 43E

Radioactive decay. A cesium isotope has a half-life of 30 years. What is the continuous compound rate of decay? (Use the radioactive decay model in Problem 43.)

43. Radioactive decay. A mathematical model for the decay of radioactive substances is given by

  $Q=Q_0{e}^{rt}$

  where

  $\begin{array}{lll}{Q}_0\hfill & =\hfill & \text{amount}\text{of}\text{the}\text{substance}\text{at}\text{time}t = 0\hfill \\ r\hfill & =\hfill & \text{continuous}\text{compound}\text{rate}\text{of}\text{decay}\hfill \\ t\hfill & =\hfill & \text{time}\text{in}\text{years}\hfill \\ Q\hfill & =\hfill & \text{amount}\text{of}\text{the}\text{substance}\text{at}\text{time}t\hfill \end{array}$

If the continuous compound rate of decay of radium per year is r = −0.000 433 2, how long will it take a certain amount of radium to decay to half the original amount? (This period is the half-life of the substance.)