Problem $2$: Pollution begins to enter a lake at time $t=0$ at a rate $($in gallons per hour$)$ given by

$f(t)=10(1-e^{-0\mathrm{.}5t})$

where $t$ is the time $($in hours$).$ At the same time, the pollution filter begins to remove the pollution at a rate of

$g(t)=0\mathrm{.}4t$

as long as the pollution remains in the lake.

a$.$ How much pollution is in the lake after $12$ hours?

b$.$ Find the time when the rate at which the pollution enters the lake equals the rate at which the pollution is removed $($use a graphing device to help you! There are two solutions$).$