Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. (Note: The volume $V$ of a right circular cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)
(a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute?
(b) A recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is $R ( t ) = 400 \sqrt { t }$ cubic centimeters per minute, where $t$ is the time in minutes since the device began working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time $t$ when the oil slick reaches its maximum volume. Justify your answer.
(c) By the time the recovery device began removing oil, 60,000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).

A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the function $S$, where $S(t)$ is measured in millimeters and $t$ is measured in days for $0 \leq t \leq 60$. The rate at which the height of the water is rising in the can is given by $S^{\prime}(t) = 2\sin(0.03t) + 1.5$.
(a) According to the model, what is the height of the water in the can at the end of the 60-day period?
(b) According to the model, what is the average rate of change in the height of water in the can over the 60-day period? Show the computations that lead to your answer. Indicate units of measure.
(c) Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time $t = 7$? Indicate units of measure.
(d) During the same 60-day period, rain on Monsoon Mountain accumulates in a can identical to the one in Stormville. The height of the water in the can on Monsoon Mountain is modeled by the function $M$, where $M(t) = \frac{1}{400}\left(3t^3 - 30t^2 + 330t\right)$. The height $M(t)$ is measured in millimeters, and $t$ is measured in days for $0 \leq t \leq 60$. Let $D(t) = M^{\prime}(t) - S^{\prime}(t)$. Apply the Intermediate Value Theorem to the function $D$ on the interval $0 \leq t \leq 60$ to justify that there exists a time $t$, $0 < t < 60$, at which the heights of water in the two cans are changing at the same rate.

On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.

On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.

The rate at which rainwater flows into a drainpipe is modeled by the function $R$, where $R ( t ) = 20 \sin \left( \frac { t ^ { 2 } } { 35 } \right)$ cubic feet per hour, $t$ is measured in hours, and $0 \leq t \leq 8$. The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by $D ( t ) = - 0.04 t ^ { 3 } + 0.4 t ^ { 2 } + 0.96 t$ cubic feet per hour, for $0 \leq t \leq 8$. There are 30 cubic feet of water in the pipe at time $t = 0$.
(a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval $0 \leq t \leq 8$?
(b) Is the amount of water in the pipe increasing or decreasing at time $t = 3$ hours? Give a reason for your answer.
(c) At what time $t$, $0 \leq t \leq 8$, is the amount of water in the pipe at a minimum? Justify your answer.
(d) The pipe can hold 50 cubic feet of water before overflowing. For $t > 8$, water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time $w$ when the pipe will begin to overflow.

Water is pumped into a tank at a rate modeled by $W ( t ) = 2000 e ^ { - t ^ { 2 } / 20 }$ liters per hour for $0 \leq t \leq 8$, where $t$ is measured in hours. Water is removed from the tank at a rate modeled by $R ( t )$ liters per hour, where $R$ is differentiable and decreasing on $0 \leq t \leq 8$. Selected values of $R ( t )$ are shown in the table below. At time $t = 0$, there are 50,000 liters of water in the tank.

\begin{tabular}{ c } $t$

(hours)

& 0 & 1 & 3 & 6 & 8 \hline

$R ( t )$

(liters / hour)

& 1340 & 1190 & 950 & 740 & 700 \hline \end{tabular}
(a) Estimate $R ^ { \prime } ( 2 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of water removed from the tank during the 8 hours. Is this an overestimate or an underestimate of the total amount of water removed? Give a reason for your answer.
(c) Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours.
(d) For $0 \leq t \leq 8$, is there a time $t$ when the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank? Explain why or why not.