Question

The half-life of cesium-137 is 30 years. Suppose we have a
100-mg sample.
(a) Find the mass that remains after years.
(b) How much of the sample remains after 100 years?
(c) After how long will only 1 mg remain?

Answer 1

(a) If $y(t)$ is the mass (in mg) remaining after $t$ years, then $y(t) = y(0) e^{kt}=100e^{kt}.$

$y(30) = 100 e^{30k} = \frac{1}{2}(100) \implies e^{30k} = \frac{1}{2} \implies k = -(\ln 2) /30 \implies \\ \\ y(t) = 100e^{-(\ln 2)t/30} = 100 \cdot 2^{-t/30}$

(b) $y(100) = 100 \cdot 2^{-100/30} \approx \text{9.92 mg}$

(c)
$100 e^{- (\ln 2)t/30} = 1\ \implies\ -(\ln 2) t / 30 = \ln \frac{1}{100}\ \implies\ \\ \\ t = -30 \frac{\ln 0.01}{\ln 2} \approx \text{199.3 years}$