# Question ab040

Sep 11, 2015

The answer is (B) $\text{11520 years}$

#### Explanation:

A radioactive isotope's half-life tells you the time needed for an initial sample of said isotope to be halved.

More specifically, an initial sample of a radioactive element will be halved for every half-life that passes. If you start at $t = 0$ and have ${t}_{\text{1/2}}$ as the isotope's half-life, then an initial sample will change like this

100% -> t = 0

50% -> t = t_"1/2"

25% -> t = 2 * t_"1/2"

12.5% -> t = 3 * t_"1/2"

6.25% -> t = 4 * t_"1/2"

This is equivalent to saying that

$\text{what you have" = "what you started with"/2^n" }$, where

$n$ - the number of half-lives that passed.

In your case, you start with a sample of 100 mg. In order for the sample to be reduced to 25 mg, you need to have

$\text{100 mg} \to t = 0$

$\text{50 mg" -> t = t_"1/2}$

$\text{25 mg" -> t = 2 * t_"1/2}$

Your remaining sample is now a quarter the size it was in the beginning, which can only mean that two half-lives have passed

$\frac{\text{25 mg" = "100 mg}}{2} ^ n$

${2}^{n} = \left(100 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{mg"))))/(25color(red)(cancel(color(black)("mg}}}}\right) = 4 \implies n = \textcolor{g r e e n}{2}$

The time that passed is thus equal to

t = 2 * t_"1/2" = 2 * "5760 years" = color(green)("11520 years")#