# The half-life of strontium-90 is 28 years. How long will it take a 44 mg sample to decay to a mass of 11 mg?

Oct 20, 2015

$\text{56 years}$

#### Explanation:

A radioactive isotope's nuclear half-life tells you how much time must pass until an initial sample is halved.

In your case, strontium-90 is known to have a half-life of $28$ years. THis means that if you start with an initial mass of strontium-90, let's say ${A}_{0}$, you will have

• ${A}_{0} \cdot \frac{1}{2} \to$ after one half-life passes;
• ${A}_{0} / 2 \cdot \frac{1}{2} = {A}_{0} / 4 \to$ after two half-lives pass;
• ${A}_{0} / 4 \cdot \frac{1}{2} = {A}_{0} / 8 \to$ after three half-lives pass;
• ${A}_{0} / 8 \cdot \frac{1}{2} = {A}_{0} / 16 \to$ after four half-lives pass;
$\vdots$

and so on.

Notice that you can write the remaining amount of an initial sample by using the number of half-lives that pass

$\text{remaining amount" = "initial amount"/2^n" }$, where

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

Now, you initial sample has a mass of $\text{44 mg}$. Notice that the remaining sample can be written as

$\frac{\text{11 mg" = "44 mg"/4 = "44 mg}}{2} ^ 2$

This means that two half-lives must pass in order for the strontium-90 sample to decay to a quarter of its initial mass.

This implies that you have

2color(red)(cancel(color(black)("half-lives"))) * "28 years"/(1color(red)(cancel(color(black)("half-life")))) = color(green)("56 years")