# A radioactive isotope has a half-life of 9 hours. How do you find the amount of the isotope left from a 80-milligram sample after 54 hours?

Dec 30, 2015

After 54 hours, 1.25 milligrams of the sample will be left.

#### Explanation:

After a half-life period, the quantity of the sample with be reduced to half.

So, after 9 hours, amount of sample left
$= 80 \cdot \frac{1}{2} = 40$ milligrams

After another 9 hours, amount of sample left
$= 40 \cdot \frac{1}{2}$
or, $80 \cdot \frac{1}{2} \cdot \frac{1}{2} = 80 \cdot {\left(\frac{1}{2}\right)}^{2} = 20$ milligrams

Do you see the pattern emerging?

In 54 hours, there are $\frac{54}{9} = 6$ half-life periods.

Therefore, amount of sample left after 54 hours
= amount of sample left after 6 half-life periods

$= 80 \cdot {\left(\frac{1}{2}\right)}^{6} = \frac{80}{64} = 1.25$ milligrams