Solve the problem.
-The rate of water usage for a business, in gallons per hour, is given by $W ( t ) = 16 t e ^ { - t }$, where $t$ is the number of hours since midnight. Find the average rate of water usage over the interval $0 \leq t \leq 5$.
A) 0.13 gallons per hour
B) 3.07 gallons per hour
C) 3.11 gallons per hour
D) 3.33 gallons per hour

Question

Quizplus · Accepted Answer

The average rate of water usage over the interval $0 \leq t \leq 5$ can be found by integrating the rate function $W(t) = 16te^{-t}$ from $t = 0$ to $t = 5$ and then dividing by the length of the interval, which is 5 hours. The integral of $16te^{-t}$ from 0 to 5 gives the total amount of water used over the 5 hours. The formula for the average value of a function $f(t)$ over an interval $a \leq t \leq b$ is $\frac{1}{b-a} \int_{a}^{b} f(t) dt$.$$\text{Average rate} = \frac{1}{5-0} \int_{0}^{5} 16te^{-t} dt$$Solving this integral and then dividing by 5 gives the average rate of water usage over the given time period. The exact calculation involves integration by parts and yields an average rate of approximately 3.07 gallons per hour.

Solve the Problem W(t)=16te−tW ( t ) = 16 t e ^ { - t }W(t)=16te−t

Solve the Problem $W ( t ) = 16 t e ^ { - t }$