$4\mathrm{.}5$ The system shown in the figure might be used to load ore onto barges

for shipment. Let $W_1=$ flow rate of ore to barge, tons$/$hour$,$ and $M=$

mass of ore at any time; tons. After the pile has reached a mass $M_0,$

ore is lost over the side of the barge at a rate $W_2$ tons$/$hour$.$ The loss

rate is about proportional to the amount of ore in excess of $M_0$ at any

time.

$($a$)$ Derive the mathematical description for this situation.

$($b$)$ Express the time required for the ore on the barge to reach a mass

$M_0$ in terms of the constant flow rate $W_1.$

$($c$)$ Derive an algebraic relationship showing $M$ as a function of $t$

for $M>M_0.$ What happens as $t?$