# The decay rate of a certain chemical is 9.8% per year what is the half life?

Jul 31, 2016

Approximately $6.72$ years

#### Explanation:

The proportion left at the end of one year is:

$\frac{100 - 9.8}{100} = \frac{90.2}{100} = 0.902$

The proportion left at the end of $t$ years will be:

${0.902}^{t}$

To find the half life we want to find $t$ such that:

${0.902}^{t} = 0.5$

Taking logs of both sides, we have:

$t \log \left(0.902\right) = \log \left({0.902}^{t}\right) = \log \left(0.5\right)$

Hence $t = \log \frac{0.5}{\log} \left(0.902\right) \approx 6.72$ years