3. The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time $t = 0$. During the time interval $0 \leq t \leq 12$ hours, water is pumped into the pool at the rate $P ( t )$ cubic feet per hour. The table above gives values of $P ( t )$ for selected values of $t$. During the same time interval, water is leaking from the pool at the rate $R ( t )$ cubic feet per hour, where $R ( t ) = 25 e ^ { - 0.05 t }$. (Note: The volume $V$ of a cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.) (a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval $0 \leq t \leq 12$ hours. Show the computations that lead to your answer. (b) Calculate the total amount of water that leaked out of the pool during the time interval $0 \leq t \leq 12$ hours. (c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time $t = 12$ hours. Round your answer to the nearest cubic foot. (d) Find the rate at which the volume of water in the pool is increasing at time $t = 8$ hours. How fast is the water level in the pool rising at $t = 8$ hours? Indicate units of measure in both answers.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
No calculator is allowed for these problems. [Figure]
3. The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time $t = 0$. During the time interval $0 \leq t \leq 12$ hours, water is pumped into the pool at the rate $P ( t )$ cubic feet per hour. The table above gives values of $P ( t )$ for selected values of $t$. During the same time interval, water is leaking from the pool at the rate $R ( t )$ cubic feet per hour, where $R ( t ) = 25 e ^ { - 0.05 t }$. (Note: The volume $V$ of a cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)\\
(a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval $0 \leq t \leq 12$ hours. Show the computations that lead to your answer.\\
(b) Calculate the total amount of water that leaked out of the pool during the time interval $0 \leq t \leq 12$ hours.\\
(c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time $t = 12$ hours. Round your answer to the nearest cubic foot.\\
(d) Find the rate at which the volume of water in the pool is increasing at time $t = 8$ hours. How fast is the water level in the pool rising at $t = 8$ hours? Indicate units of measure in both answers.
\section*{WRITE ALL WORK IN THE EXAM BOOKLET.}
\section*{END OF PART A OF SECTION II}
No calculator is allowed for these problems.\\
\includegraphics[max width=\textwidth, alt={}, center]{d04a7a11-0c12-4aac-bb2f-8c47963b3289-4_637_1309_586_408}\\