# Half-life is defined as the amount of time that it takes for radioactive substance to loose half its radioactivity. If a substance has a half life of 58 years and starts with 500 g radioactive, how much remains radioactive after 30 years?

Jan 27, 2017

You can work out a yearly decay-factor, which is just a growth factor (known from exponential functions in maths or economy), but now it's below 1.

#### Explanation:

$N = B \times {G}^{t}$, where N=new value, B=start value, G=growth (or decay) factor, and t=number of periods (years in this case.

We know that ${G}^{58} = 1 / 2$, so
$G = \sqrt[58]{\frac{1}{2}} = {\left(\frac{1}{2}\right)}^{\frac{1}{58}}$

After 30 years, activity will be:

$N = 500 \times {\left(\sqrt[58]{\frac{1}{2}}\right)}^{30} = 500 \times {\left(\frac{1}{2}\right)}^{\frac{30}{58}} = 500 \times 0.699 = 349 g$

On your calculator you can set this up like:

0.5 ^ (30 : 58) x 500 = 349