Question

Element X decays radioactively with a half life of 9 minutes. If there are 610grams of

Element X, how long, to the nearest tenth of a minute, would it take the element to
decay to 124 grams?
y = a(.5) t/h

Answer 1

A half-life of 9 minutes corresponds to a decay factor k such that

`\frac12=e^(9k)\implies k=-\frac19\ln\left(\frac12\right)`

Then the time it takes for the substance to decay from 610 g to 124 g is time t such that

`124\,\mathrm g=(610\,\mathrm g)e^(kt)`

Solve for t :

`(62)/(305)=e^(kt)`

`\ln\left((62)/(305)\right)=kt`

`t=\frac1k\ln\left((62)/(305)\right)\approx20.686`

so it takes about 20.7 min for the 610 g sample to decay down to 124 g.