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.

A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the edge of the water was changing during the storm is modeled by $f ( t ) = \sqrt { t } + \cos t - 3$ meters per hour, $t$ hours after the storm began. The edge of the water was 35 meters from the road when the storm began, and the storm lasted 5 hours. The derivative of $f ( t )$ is $f ^ { \prime } ( t ) = \frac { 1 } { 2 \sqrt { t } } - \sin t$.
(a) What was the distance between the road and the edge of the water at the end of the storm?
(b) Using correct units, interpret the value $f ^ { \prime } ( 4 ) = 1.007$ in terms of the distance between the road and the edge of the water.
(c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water decreasing most rapidly? Justify your answer.
(d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the edge of the water was growing at a rate of $g ( p )$ meters per day, where $p$ is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water.

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 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 $S ^ { \prime } ( t ) = 2 \sin ( 0.03 t ) + 1.5$.
(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 $M ( t ) = \frac { 1 } { 400 } \left( 3 t ^ { 3 } - 30 t ^ { 2 } + 330 t \right)$. 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.

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.

A water tank in the form of a right rectangular parallelepiped, with 4 m in length, 3 m in width, and 2 m in height, needs to be sanitized. In this operation, the tank will need to be emptied in 20 minutes at most. The water will be removed with the help of a pump with constant flow rate, where flow rate is the volume of liquid that passes through the pump per unit of time.
The minimum flow rate, in liters per second, that this pump should have so that the tank is emptied in the stipulated time is
(A) 2.
(B) 3.
(C) 5.
(D) 12.
(E) 20.

A blood bank receives 450 mL of blood from each donor. After separating blood plasma from red blood cells, the former is stored in bags with 250 mL capacity. The blood bank rents refrigerators from a company for storage of plasma bags, according to its needs. Each refrigerator has a storage capacity of 50 bags. Over the course of a week, 100 people donated blood to that bank.
Assume that from every 60 mL of blood, 40 mL of plasma is extracted.
The minimum number of freezers that the bank needed to rent to store all the plasma bags from that week was
(A) 2.
(B) 3.
(C) 4.
(D) 6.
(E) 8.

The package of snacks preferred by a girl is sold in packages with different quantities. Each package is assigned a number of points in the promotion: ``When you total exactly 12 points in packages and add another R\$ 10.00 to the purchase value, you will win a stuffed animal''.
This snack is sold in three packages with the following masses, points, and prices:

\begin{tabular}{ c } Package

mass (g)

&

Package

points

& Price (R\$) \hline 50 & 2 & 2.00 \hline 100 & 4 & 3.60 \hline 200 & 6 & 6.40 \hline \end{tabular}
The smallest amount to be spent by this girl that allows her to take the stuffed animal in this promotion is
(A) R\$ 10.80.
(B) R\$ 12.80.
(C) R\$ 20.80.
(D) R\$ 22.00.
(E) R\$ 22.80.

The venue for Olympic swimming competitions uses the most advanced technology to provide swimmers with ideal conditions. This involves reducing the impact of undulation and currents caused by swimmers in their movement. To achieve this, the competition pool has a uniform depth of 3 m, which helps reduce the ``reflection'' of water (the movement against a surface and the return in the opposite direction, reaching the swimmers), in addition to the already traditional 50 m length and 25 m width. A club wishes to reform its pool of 50 m length, 20 m width and 2 m depth so that it has the same dimensions as Olympic pools.
After the reform, the capacity of this pool will exceed the capacity of the original pool by a value closest to
(A) $20\%$.
(B) $25\%$.
(C) $47\%$.
(D) $50\%$.
(E) $88\%$.

A couple is moving to a new home and needs to place a cubic object, with 80 cm edges, in a cardboard box, which cannot be disassembled. They have five boxes available, with different dimensions, as described:

• Box 1: $86 \mathrm{~cm} \times 86 \mathrm{~cm} \times 86 \mathrm{~cm}$
• Box 2: $75 \mathrm{~cm} \times 82 \mathrm{~cm} \times 90 \mathrm{~cm}$
• Box 3: $85 \mathrm{~cm} \times 82 \mathrm{~cm} \times 90 \mathrm{~cm}$
• Box 4: $82 \mathrm{~cm} \times 95 \mathrm{~cm} \times 82 \mathrm{~cm}$
• Box 5: $80 \mathrm{~cm} \times 95 \mathrm{~cm} \times 85 \mathrm{~cm}$

The couple needs to choose a box in which the object fits, so that the least free space remains in its interior.
The box chosen by the couple should be number
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 5.

The following is a graph showing the velocity $v ( t )$ at time $t$ ( $0 \leqq t \leqq d$ ) of a point P moving on a number line starting from the origin.
When $\int _ { 0 } ^ { a } | v ( t ) | d t = \int _ { a } ^ { d } | v ( t ) | d t$, which of the following statements in are correct? (Here, $0 < a < b < c < d$.) [3 points]
Remarks ㄱ. Point P passes through the origin again after starting. ㄴ. $\int _ { 0 } ^ { c } v ( t ) d t = \int _ { c } ^ { d } v ( t ) d t$ ㄷ. $\int _ { 0 } ^ { b } v ( t ) d t = \int _ { b } ^ { d } | v ( t ) | d t$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The position $x ( t )$ of a point P moving on a number line at time $t$ is given by $$x ( t ) = t ( t - 1 ) ( a t + b ) \quad ( a \neq 0 )$$ for two constants $a , b$. The velocity $v ( t )$ of point P at time $t$ satisfies $\int _ { 0 } ^ { 1 } | v ( t ) | d t = 2$. Which of the following statements in the given options are correct? [4 points]
Given statements: ᄀ. $\int _ { 0 } ^ { 1 } v ( t ) d t = 0$ ㄴ. There exists $t _ { 1 }$ in the open interval $( 0,1 )$ such that $\left| x \left( t _ { 1 } \right) \right| > 1$. ㄷ. If $| x ( t ) | < 1$ for all $t$ with $0 \leq t \leq 1$, then there exists $t _ { 2 }$ in the open interval $( 0,1 )$ such that $x \left( t _ { 2 } \right) = 0$.
(1) ᄀ
(2) ᄀ, ㄴ
(3) ᄀ, ㄷ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes