Honey is poured through a funnel at a rate of $r ( t ) = 4 e ^ { - 0.35 t }$ ounces per minute, where $t$ is measured in minutes. How many ounces of honey are poured through the funnel from time $t = 0$ to time $t = 3$?
(A) 0.910
(B) 1.400
(C) 2.600
(D) 7.429

The rate at which people enter an amusement park on a given day is modeled by the function $E$ defined by $$E ( t ) = \frac { 15600 } { \left( t ^ { 2 } - 24 t + 160 \right) }$$ The rate at which people leave the same amusement park on the same day is modeled by the function $L$ defined by $$L ( t ) = \frac { 9890 } { \left( t ^ { 2 } - 38 t + 370 \right) }$$ Both $E ( t )$ and $L ( t )$ are measured in people per hour and time $t$ is measured in hours after midnight. These functions are valid for $9 \leq t \leq 23$, the hours during which the park is open. At time $t = 9$, there are no people in the park.
(a) How many people have entered the park by 5:00 P.M. ( $t = 17$ )? Round your answer to the nearest whole number.
(b) The price of admission to the park is $\$15$ until 5:00 P.M. ( $t = 17$ ). After 5:00 P.M., the price of admission to the park is $\$11$. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.
(c) Let $H ( t ) = \int _ { 9 } ^ { t } ( E ( x ) - L ( x ) ) d x$ for $9 \leq t \leq 23$. The value of $H ( 17 )$ to the nearest whole number is 3725. Find the value of $H ^ { \prime } ( 17 )$, and explain the meaning of $H ( 17 )$ and $H ^ { \prime } ( 17 )$ in the context of the amusement park.
(d) At what time $t$, for $9 \leq t \leq 23$, does the model predict that the number of people in the park is a maximum?

For $0 \leq t \leq 31$, the rate of change of the number of mosquitoes on Tropical Island at time $t$ days is modeled by $R(t) = 5\sqrt{t}\cos\left(\frac{t}{5}\right)$ mosquitoes per day. There are 1000 mosquitoes on Tropical Island at time $t = 0$.
(a) Show that the number of mosquitoes is increasing at time $t = 6$.
(b) At time $t = 6$, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer.
(c) According to the model, how many mosquitoes will be on the island at time $t = 31$? Round your answer to the nearest whole number.
(d) To the nearest whole number, what is the maximum number of mosquitoes for $0 \leq t \leq 31$? Show the analysis that leads to your conclusion.

The tide removes sand from Sandy Point Beach at a rate modeled by the function $R$, given by $$R ( t ) = 2 + 5 \sin \left( \frac { 4 \pi t } { 25 } \right)$$ A pumping station adds sand to the beach at a rate modeled by the function $S$, given by $$S ( t ) = \frac { 15 t } { 1 + 3 t }$$ Both $R ( t )$ and $S ( t )$ have units of cubic yards per hour and $t$ is measured in hours for $0 \leq t \leq 6$. At time $t = 0$, the beach contains 2500 cubic yards of sand.
(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.
(b) Write an expression for $Y ( t )$, the total number of cubic yards of sand on the beach at time $t$.
(c) Find the rate at which the total amount of sand on the beach is changing at time $t = 4$.
(d) For $0 \leq t \leq 6$, at what time $t$ is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.

A water tank at Camp Newton holds 1200 gallons of water at time $t = 0$. During the time interval $0 \leq t \leq 18$ hours, water is pumped into the tank at the rate $$W(t) = 95\sqrt{t}\sin^2\left(\frac{t}{6}\right) \text{ gallons per hour.}$$ During the same time interval, water is removed from the tank at the rate $$R(t) = 275\sin^2\left(\frac{t}{3}\right) \text{ gallons per hour.}$$
(a) Is the amount of water in the tank increasing at time $t = 15$? Why or why not?
(b) To the nearest whole number, how many gallons of water are in the tank at time $t = 18$?
(c) At what time $t$, for $0 \leq t \leq 18$, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.
(d) For $t > 18$, no water is pumped into the tank, but water continues to be removed at the rate $R(t)$ until the tank becomes empty. Let $k$ be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.

At an intersection in Thomasville, Oregon, cars turn left at the rate $L(t) = 60\sqrt{t}\sin^{2}\left(\frac{t}{3}\right)$ cars per hour over the time interval $0 \leq t \leq 18$ hours. The graph of $y = L(t)$ is shown above.
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval $0 \leq t \leq 18$ hours.
(b) Traffic engineers will consider turn restrictions when $L(t) \geq 150$ cars per hour. Find all values of $t$ for which $L(t) \geq 150$ and compute the average value of $L$ over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval $0 \leq t \leq 7$, where $t$ is measured in hours. In this model, rates are given as follows:
(i) The rate at which water enters the tank is $f(t) = 100t^{2}\sin(\sqrt{t})$ gallons per hour for $0 \leq t \leq 7$.
(ii) The rate at which water leaves the tank is $$g(t) = \left\{ \begin{array}{r} 250 \text{ for } 0 \leq t < 3 \\ 2000 \text{ for } 3 < t \leq 7 \end{array} \right. \text{ gallons per hour.}$$ The graphs of $f$ and $g$, which intersect at $t = 1.617$ and $t = 5.076$, are shown in the figure above. At time $t = 0$, the amount of water in the tank is 5000 gallons.
(a) How many gallons of water enter the tank during the time interval $0 \leq t \leq 7$? Round your answer to the nearest gallon.
(b) For $0 \leq t \leq 7$, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.
(c) For $0 \leq t \leq 7$, at what time $t$ is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.

The rate at which people enter an auditorium for a rock concert is modeled by the function $R$ given by $R(t) = 1380t^{2} - 675t^{3}$ for $0 \leq t \leq 2$ hours; $R(t)$ is measured in people per hour. No one is in the auditorium at time $t = 0$, when the doors open. The doors close and the concert begins at time $t = 2$.
(a) How many people are in the auditorium when the concert begins?
(b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer.
(c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function $w$ models the total wait time for all the people who enter the auditorium before time $t$. The derivative of $w$ is given by $w'(t) = (2 - t)R(t)$. Find $w(2) - w(1)$, the total wait time for those who enter the auditorium after time $t = 1$.
(d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).

Mighty Cable Company manufactures cable that sells for $\$120$ per meter. For a cable of fixed length, the cost of producing a portion of the cable varies with its distance from the beginning of the cable. Mighty reports that the cost to produce a portion of a cable that is $x$ meters from the beginning of the cable is $6\sqrt{x}$ dollars per meter. (Note: Profit is defined to be the difference between the amount of money received by the company for selling the cable and the company's cost of producing the cable.)
(a) Find Mighty's profit on the sale of a 25-meter cable.
(b) Using correct units, explain the meaning of $\int_{25}^{30} 6\sqrt{x} \, dx$ in the context of this problem.
(c) Write an expression, involving an integral, that represents Mighty's profit on the sale of a cable that is $k$ meters long.
(d) Find the maximum profit that Mighty could earn on the sale of one cable. Justify your answer.

There is no snow on Janet's driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by $f(t) = 7te^{\cos t}$ cubic feet per hour, where $t$ is measured in hours since midnight. Janet starts removing snow at 6 A.M. ($t = 6$). The rate $g(t)$, in cubic feet per hour, at which Janet removes snow from the driveway at time $t$ hours after midnight is modeled by $$g(t) = \begin{cases} 0 & \text{for } 0 \leq t < 6 \\ 125 & \text{for } 6 \leq t < 7 \\ 108 & \text{for } 7 \leq t \leq 9. \end{cases}$$
(a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.?
(b) Find the rate of change of the volume of snow on the driveway at 8 A.M.
(c) Let $h(t)$ represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time $t$ hours after midnight. Express $h$ as a piecewise-defined function with domain $0 \leq t \leq 9$.
(d) How many cubic feet of snow are on the driveway at 9 A.M.?

There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later. While there is a line, people move onto the ride at a rate of 800 people per hour. The graph above shows the rate, $r(t)$, at which people arrive at the ride throughout the day. Time $t$ is measured in hours from the time the ride begins operation.
(a) How many people arrive at the ride between $t = 0$ and $t = 3$? Show the computations that lead to your answer.
(b) Is the number of people waiting in line to get on the ride increasing or decreasing between $t = 2$ and $t = 3$? Justify your answer.
(c) At what time $t$ is the line for the ride the longest? How many people are in line at that time? Justify your answers.
(d) Write, but do not solve, an equation involving an integral expression of $r$ whose solution gives the earliest time $t$ at which there is no longer a line for the ride.

A 12,000-liter tank of water is filled to capacity. At time $t = 0$, water begins to drain out of the tank at a rate modeled by $r(t)$, measured in liters per hour, where $r$ is given by the piecewise-defined function $$r(t) = \begin{cases} \dfrac{600t}{t+3} & \text{for } 0 \leq t \leq 5 \\ 1000e^{-0.2t} & \text{for } t > 5 \end{cases}$$
(a) Is $r$ continuous at $t = 5$? Show the work that leads to your answer.
(b) Find the average rate at which water is draining from the tank between time $t = 0$ and time $t = 8$ hours.
(c) Find $r^{\prime}(3)$. Using correct units, explain the meaning of that value in the context of this problem.
(d) Write, but do not solve, an equation involving an integral to find the time $A$ when the amount of water in the tank is 9000 liters.

Water is pumped into a tank at a rate of $r ( t ) = 30 \left( 1 - e ^ { - 0.16 t } \right)$ gallons per minute, where $t$ is the number of minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 20 minutes?
(A) 380 gallons
(B) 420 gallons
(C) 829 gallons
(D) 1220 gallons
(E) 1376 gallons

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 $D(t) = -0.04t^3 + 0.4t^2 + 0.96t$ 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.

When a certain grocery store opens, it has 50 pounds of bananas on a display table. Customers remove bananas from the display table at a rate modeled by $$f(t) = 10 + (0.8t)\sin\left(\frac{t^3}{100}\right) \text{ for } 0 < t \leq 12$$ where $f(t)$ is measured in pounds per hour and $t$ is the number of hours after the store opened. After the store has been open for three hours, store employees add bananas to the display table at a rate modeled by $$g(t) = 3 + 2.4\ln\left(t^2 + 2t\right) \text{ for } 3 < t \leq 12$$ where $g(t)$ is measured in pounds per hour and $t$ is the number of hours after the store opened.
(a) How many pounds of bananas are removed from the display table during the first 2 hours the store is open?
(b) Find $f'(7)$. Using correct units, explain the meaning of $f'(7)$ in the context of the problem.
(c) Is the number of pounds of bananas on the display table increasing or decreasing at time $t = 5$? Give a reason for your answer.
(d) How many pounds of bananas are on the display table at time $t = 8$?

People enter a line for an escalator at a rate modeled by the function $r$ given by
$$r ( t ) = \begin{cases} 44 \left( \frac { t } { 100 } \right) ^ { 3 } \left( 1 - \frac { t } { 300 } \right) ^ { 7 } & \text { for } 0 \leq t \leq 300 \\ 0 & \text { for } t > 300 \end{cases}$$
where $r ( t )$ is measured in people per second and $t$ is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time $t = 0$.
(a) How many people enter the line for the escalator during the time interval $0 \leq t \leq 300$ ?
(b) During the time interval $0 \leq t \leq 300$, there are always people in line for the escalator. How many people are in line at time $t = 300$ ?
(c) For $t > 300$, what is the first time $t$ that there are no people in line for the escalator?
(d) For $0 \leq t \leq 300$, at what time $t$ is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify your answer.

Fish enter a lake at a rate modeled by the function $E$ given by $E(t) = 20 + 15\sin\left(\frac{\pi t}{6}\right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L(t) = 4 + 2^{0.1t^2}$. Both $E(t)$ and $L(t)$ are measured in fish per hour, and $t$ is measured in hours since midnight $(t = 0)$.
(a) How many fish enter the lake over the 5-hour period from midnight $(t = 0)$ to 5 A.M. $(t = 5)$? Give your answer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $(t = 0)$ to 5 A.M. $(t = 5)$?
(c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 a.m. $(t = 5)$? Explain your reasoning.

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A(t) = 450\sqrt{\sin(0.62t)}$, where $t$ is the number of hours after 5 A.M. and $A(t)$ is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.
(a) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. $(t=1)$ to 10 A.M. $(t=5)$.
(b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. $(t=1)$ to 10 A.M. $(t=5)$.
(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ($t=1$) increasing or decreasing? Give a reason for your answer.
(d) A line forms whenever $A(t) \geq 400$. The number of vehicles in line at time $t$, for $a \leq t \leq 4$, is given by $N(t) = \int_{a}^{t}(A(x) - 400)\,dx$, where $a$ is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval $a \leq t \leq 4$. Justify your answer.

The rate at which people enter an amusement park on a given day is modeled by the function $E$ defined by $$E ( t ) = \frac { 15600 } { \left( t ^ { 2 } - 24 t + 160 \right) }$$ The rate at which people leave the same amusement park on the same day is modeled by the function $L$ defined by $$L ( t ) = \frac { 9890 } { \left( t ^ { 2 } - 38 t + 370 \right) }$$ Both $E ( t )$ and $L ( t )$ are measured in people per hour and time $t$ is measured in hours after midnight. These functions are valid for $9 \leq t \leq 23$, the hours during which the park is open. At time $t = 9$, there are no people in the park.
(a) How many people have entered the park by 5:00 P.M. ( $t = 17$ )? Round your answer to the nearest whole number.
(b) The price of admission to the park is $\$15$ until 5:00 P.M. ( $t = 17$ ). After 5:00 P.M., the price of admission to the park is $\$11$. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.
(c) Let $H ( t ) = \int _ { 9 } ^ { t } ( E ( x ) - L ( x ) ) d x$ for $9 \leq t \leq 23$. The value of $H ( 17 )$ to the nearest whole number is 3725. Find the value of $H ^ { \prime } ( 17 )$, and explain the meaning of $H ( 17 )$ and $H ^ { \prime } ( 17 )$ in the context of the amusement park.
(d) At what time $t$, for $9 \leq t \leq 23$, does the model predict that the number of people in the park is a maximum?

At an intersection in Thomasville, Oregon, cars turn left at the rate $L(t) = 60\sqrt{t}\sin^{2}\left(\frac{t}{3}\right)$ cars per hour over the time interval $0 \leq t \leq 18$ hours.
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval $0 \leq t \leq 18$ hours.
(b) Traffic engineers will consider turn restrictions when $L(t) \geq 150$ cars per hour. Find all values of $t$ for which $L(t) \geq 150$ and compute the average value of $L$ over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval $0 \leq t \leq 7$, where $t$ is measured in hours. In this model, rates are given as follows:
(i) The rate at which water enters the tank is $f(t) = 100t^2 \sin(\sqrt{t})$ gallons per hour for $0 \leq t \leq 7$.
(ii) The rate at which water leaves the tank is $$g(t) = \left\{ \begin{array}{r} 250 \text{ for } 0 \leq t < 3 \\ 2000 \text{ for } 3 < t \leq 7 \end{array} \right. \text{ gallons per hour.}$$
The graphs of $f$ and $g$, which intersect at $t = 1.617$ and $t = 5.076$, are shown in the figure above. At time $t = 0$, the amount of water in the tank is 5000 gallons.
(a) How many gallons of water enter the tank during the time interval $0 \leq t \leq 7$? Round your answer to the nearest gallon.
(b) For $0 \leq t \leq 7$, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.
(c) For $0 \leq t \leq 7$, at what time $t$ is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.

There is no snow on Janet's driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by $f(t) = 7te^{\cos t}$ cubic feet per hour, where $t$ is measured in hours since midnight. Janet starts removing snow at 6 A.M. $(t = 6)$. The rate $g(t)$, in cubic feet per hour, at which Janet removes snow from the driveway at time $t$ hours after midnight is modeled by $$g(t) = \begin{cases} 0 & \text{for } 0 \leq t < 6 \\ 125 & \text{for } 6 \leq t < 7 \\ 108 & \text{for } 7 \leq t \leq 9. \end{cases}$$
(a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.?
(b) Find the rate of change of the volume of snow on the driveway at 8 A.M.
(c) Let $h(t)$ represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time $t$ hours after midnight. Express $h$ as a piecewise-defined function with domain $0 \leq t \leq 9$.
(d) How many cubic feet of snow are on the driveway at 9 A.M.?

A tank contains 50 liters of oil at time $t = 4$ hours. Oil is being pumped into the tank at a rate $R ( t )$, where $R ( t )$ is measured in liters per hour, and $t$ is measured in hours. Selected values of $R ( t )$ are given in the table above. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time $t = 15$ hours?

$t$ (hours)	4	7	12	15
$R ( t )$ (liters/hour)	6.5	6.2	5.9	5.6

(A) 64.9
(B) 68.2
(C) 114.9
(D) 116.6
(E) 118.2

People enter a line for an escalator at a rate modeled by the function $r$ given by
$$r ( t ) = \begin{cases} 44 \left( \frac { t } { 100 } \right) ^ { 3 } \left( 1 - \frac { t } { 300 } \right) ^ { 7 } & \text { for } 0 \leq t \leq 300 \\ 0 & \text { for } t > 300 \end{cases}$$
where $r ( t )$ is measured in people per second and $t$ is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time $t = 0$.
(a) How many people enter the line for the escalator during the time interval $0 \leq t \leq 300$ ?
(b) During the time interval $0 \leq t \leq 300$, there are always people in line for the escalator. How many people are in line at time $t = 300$ ?
(c) For $t > 300$, what is the first time $t$ that there are no people in line for the escalator?
(d) For $0 \leq t \leq 300$, at what time $t$ is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify your answer.

Fish enter a lake at a rate modeled by the function $E$ given by $E ( t ) = 20 + 15 \sin \left( \frac { \pi t } { 6 } \right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L ( t ) = 4 + 2 ^ { 0.1 t ^ { 2 } }$. Both $E ( t )$ and $L ( t )$ are measured in fish per hour, and $t$ is measured in hours since midnight $( t = 0 )$.
(a) How many fish enter the lake over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$? Give your answer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$?
(c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. ($t = 5$)? Explain your reasoning.