Chapter 4.4, Problem 46E

Total and average daily profit. Shylls, Inc., determines that its marginal revenue per day is given by

$R^{\prime }\left(t\right)=100e^t,R\left(0\right)=0$,

where $R\left(t\right)$ is the total accumulated revenue, in dollars, on the $t\text{th}$ day. The company’s marginal cost per day is given by

$C^{\prime }\left(t\right)=100-0.2t,C\left(0\right)=0$,

where $C\left(t\right)$ is the total accumulated cost, in dollars, on the $t\text{th}$

day.

a. Find the total profit from $t=0\text{ to }t=10$

(the first 10 days).

$P\left(T\right)=R\left(T\right)-C\left(T\right)={\displaystyle {\int }_0^T\left[R\text{'}\left(t\right)-C\text{'}\left(t\right)\right]}dt$.

b. Find the average daily profit for the first 10 days $\left(\text{from }t=0\text{ to }t=10\right)$.