# Radioactive element X has a half-life of 30 days. A rock sample contains 4 grams element X when it forms. How many half-lives will have elapsed in 90 days? How much of the original amount of X will be unchanged after 90 days?

$3$ & $0.5 \setminus g m$

#### Explanation:

Given that radioactive element X has a half-life $30$ days

Hence the number $\left(n\right)$ of half-lives in $90$ days

$n = \frac{90}{30}$

$n = 3$

The amount $N$ of radioactive element X left after $n = 3$ half-lives with initial amount ${N}_{0} = 4 \setminus g m$

$N = {N}_{0} {\left(\frac{1}{2}\right)}^{n}$

$= 4 {\left(\frac{1}{2}\right)}^{3}$

$= \frac{4}{8}$

$= 0.5$

hence the amount left after $90$ days or $3$ half-lives will be $0.5$ grams