Chapter 5.2, Problem 47E

The model

${\displaystyle {\int }_0^{T}P{e}^{-kt}}dt=\frac{P}{k}\left(1-e^{-kT}\right)$

can be applied to calculate the buildup of a radioactive material that is being released into the atmosphere at a constant annual rate. Some of the material decays, but more continues to be released. The amount present at time T is given by the integral above, where P is the amount released per year and k is the half-life.

Radioactive Buildup. Cesium-137 has a decay rate of 2.3% per year. Suppose cesium-137 is released into the atmosphere for 20 yr at a rate of 1 lb per year. How much cesium-137 will be present in the atmosphere after 20 yr?