Question

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

Answer 1

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

$\dfrac12=e^{9k}\implies k=-\dfrac19\ln\left(\dfrac12\right)$

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 :

$\dfrac{62}{305}=e^{kt}$

$\ln\left(\dfrac{62}{305}\right)=kt$

$t=\dfrac1k\ln\left(\dfrac{62}{305}\right)\approx20.686$

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