Answer
$ t\approx 5.5$ hours.
Work Step by Step
Model: $ A=A_{0}e^{kt}$
Unknown: $ k $, when $ A=0.5A_{0},\ t=36$
$0.5A_{0}=A_{0}e^{k\cdot 36}\qquad... /\div A_{0}$
$0.5=e^{k\cdot 36}\qquad.../\ln(...)$
$-0.693147=36k\qquad.../\div(36)$
$ k\approx-0.019254$
So, our model is
$ A=A_{0}e^{-0.019254t}$
Now we find $ t $ for $ A=0.9A_{0}$
$0.9A_{0}=A_{0}e^{-0.019254t}\qquad... /\div A_{0}$
$0.9=e^{-0.019254t}\qquad.../\ln(...)$
$\ln 0.9 =-0.019254t\qquad.../\div(-0.019254)$
$\displaystyle \frac{\ln 0.9}{-0.019254}=t $
$ t\approx 5.5$ hours.
