Question

The half life of a radioactive element is five years. A scientist has 18 grams of the element. The equation representing the number of grams, g, after x years is g=18(0.5)$^\frac{x}{5}$ . What is the annual rate of decay?
3%
13%
87%
97%

Answer 1

Answer

It is 13%

take 2.33 x 100/18 that gives you 12.944444444 and you just round that up to 13.

Answer 2

$\bf g=18(0.5)^{\frac{x}{5}} \\\\\\ \textit{how much is there after a year?}\qquad g=18(0.5)^{\frac{1}{5}}\implies g\approx 15.67$

so, it was initially 18grams, after 1year, x = 1, it went down to 15.67grams, for a difference of about 2.33 grams.

now, if we take 18 to be the 100%, how much is 2.33 in percentage off of it?

$\bf \begin{array}{ccll} amount&\%\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 18&100\\ 2.33&p \end{array}\implies \cfrac{18}{2.33}=\cfrac{100}{p}\implies p=\cfrac{2.33\cdot 100}{18}$

round it up if you need to, that's the percentage differential.