Question

Exponential Decay and Growth - Calculus

After three days, a sample of radon-222 has decayed to 60% of its original amount. If radon-222 decays at a rate proportional to its size,

(a) find an expression for the amount of radon-222 present at any time

(b) What is the half-life of radon-222(That is, how long does it take a sample of radon-222 to decay to 50% of its original amount?)

Answer 1

the snap way to do it is to say
\[Q=Q_0(.6)^{\frac{t}{3}}\]

Answer 2

then if you want to find the half live, set
\[.6^{\frac{t}{3}}=\frac{1}{2}\] and solve for \(t\) via
\[t=\frac{3\times \ln(.5)}{\ln(.6)}\]

Answer 3

otherwise you have to go through a bunch of work to find \(r\) in the expression
\[Q=Q_0e^{rt}\] and then set it equal to \(\frac{1}{2}\) and solve

Answer 4

Thank you