Answer
$6.334~hours$
Work Step by Step
$N=175e^{kt}$
$420=175e^{k8}~~$
$\frac{420}{175}=e^{8k}$
$e^{8k}=2.4$
$\ln e^{8k}=\ln2.4$
$8k=\ln2.4$
$k=\frac{\ln2.4}{8}$
The double of 420 is 840:
$840=175e^{kt}~~$
$840=175e^{\frac{\ln2.4}{8}t}$
$\frac{840}{175}=e^{\frac{\ln2.4}{8}t}$
$4.8=e^{\frac{\ln2.4}{8}t}$
$\ln4.8=\ln e^{\frac{\ln2.4}{8}t}$
$\ln4.8=\frac{\ln2.4}{8}t$
$8\frac{\ln4.8}{\ln2.4}=t$
$t=14.334$
It takes $14.334-8=6.334~hours$ for the population to double the size.
