Chapter 8.2, Problem 31ES

Radon-222 is a radioactive gas with a half-life of 3.83 days. This gas is a health hazard because it tends to get trapped in the basements of houses, and many health officials suggest that homeowners seal their basements to prevent entry of the gas. Assume that $5.0\times {10}^7$ radon atoms are trapped in a basement at the time it is sealed and that $y\left(t\right)$ is the number of atoms present t days later.

(a) Find an initial-value problem whose solution is $y\left(t\right).$

(b) Find a formula for $y\left(t\right).$

(c) How many atoms will be present after 30 days?

(d) How long will it take for $90\%$ of the original quantity of gas to decay?