Panic escape. The figure here shows a general situation in which a stream of people attempt to escape through an exit door that turns out to be locked. The people move toward the door at speed $v_s=3\mathrm{.}350\frac{m}{s},$ are each $d=0\mathrm{.}290m$ in depth, and are separated by $L=1\mathrm{.}850m.$ The arrangement in the figure occurs at time $t=0.$ In time, people become stacked up at the door $($a$)$ at what average rate does the layer of people at the door increase? $($b$)$ At what time does the layer's depth reach $4\mathrm{.}52$ m $?($The answers reveal how quickly such a situation becomes dangerous.$)$