Question

Question #ab040

Answer 1

The answer is (B) `"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_"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

`"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

`"100 mg" -> t = 0`

`"50 mg" -> t = t_"1/2"`

`"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

`"25 mg" = "100 mg"/2^n`

`2^n = (100color(red)(cancel(color(black)("mg"))))/(25color(red)(cancel(color(black)("mg")))) = 4 implies n = color(green)(2)`

The time that passed is thus equal to

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