Question

A store is having a 12-hour sale. The rate at which shoppers enter the store, measured in shoppers per hour, is $S(t)=2 t^3-48 t^2+288 t$ for $0 \leq t \leq 12$. The rate at which shoppers leave the store, measured in shoppers per hour, is $L(t)=-80+\frac{4400}{t^2-14 t+55}$ for $0 \leq t \leq 12$. At $t=0$, when the sale begins, there are 10 shoppers in the store.
a) How many shoppers entered the store during the first six hours of the sale?
b) is the number of shoppers in the store increasing or decreasing at $t=4$ ? Explain your reasoning?
c) Find $L^{\prime}(8)$ and using correct units, explain what it means in the context of the problem.
d) Find the number of shoppers in the store, to the nearest whole number, at $t=12$.
e) At what time during the sale, $0 \leq t \leq 12$, is the number of people in the store at a maximum. Justify your answer.