The isotope of plutonium ${?}^{238}Pu$ is used to make thermoelectric power sources for spacecraft. Suppose that a space probe was launched in $2012$ with $4\mathrm{.}0kg$ of ${?}^{238}Pu.$

$($a$)$ If the half$-$life of ${?}^{238}Pu$ is $87\mathrm{.}7yr,$ write a function of the form $Q(t)=Q_0{e}^{-kt}$ to model the quantity $Q(t)$ of ${?}^{238}$ Pu left after $t$ years. Round the value of $k$ to five decimal places. Do not round intermediate calculations.

$Q(t)=$

e