Thorium-228 has a half life of 1.912 years. In how many years will 75% of a sample decompose? Round answer to nearest thousandth. Show all work.
I would let $\displaystyle A(t)$ be the amount of Thorium-228 present at time $\displaystyle t$. From the information given regarding the half-life, we may then write:
$\displaystyle A(t)=A_0\left(\frac{1}{2}\right)^{\Large \frac{t}{1.912}}$
Now, set the equal to a quarter of the initial sample:
$\displaystyle A_0\left(\frac{1}{2}\right)^{\Large \frac{t}{1.912}}=\frac{1}{4}A_0$
Divide through by $\displaystyle A_0$:
$\displaystyle \left(\frac{1}{2}\right)^{\Large \frac{t}{1.912}}= \frac{1}{4}=\left(\frac{1}{2}\right)^2$
Equate exponents:
$\displaystyle \frac{t}{1.912}=2$
$\displaystyle t=3.824\quad\checkmark$