The temperature of a room, in degrees Fahrenheit, is modeled by $H$, a differentiable function of the number of minutes after the thermostat is adjusted. Of the following, which is the best interpretation of $H ^ { \prime } ( 5 ) = 2$ ?
(A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is adjusted.
(B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes after the thermostat is adjusted.
(C) The temperature of the room is increasing at a constant rate of $\frac { 2 } { 5 }$ degree Fahrenheit per minute.
(D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute, 5 minutes after the thermostat is adjusted.

36. Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of the one with maximum volume?
(A) 3 cm
(B) 10 cm
(C) 20 cm
(D) $\frac { 30 } { \pi ^ { 2 } } \mathrm {~cm}$
(E) $\frac { 10 } { \pi } \mathrm {~cm}$

1993 AP Calculus BC: Section I

1. If $f ( x ) = \left\{ \begin{array} { l l } x & \text { for } x \leq 1 \\ \frac { 1 } { x } & \text { for } x > 1 , \end{array} \right.$ then $\int _ { 0 } ^ { e } f ( x ) d x =$
   (A) 0
   (B) $\frac { 3 } { 2 }$
   (C) 2
   (D) $e$
   (E) $e + \frac { 1 } { 2 }$
2. During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, how many people are infected 12 days after the epidemic is first discovered?
   (A) 343
   (B) 1,343
   (C) 1,367
   (D) 1,400
   (E) 2,057
3. If $\frac { d y } { d x } = \frac { 1 } { x }$, then the average rate of change of $y$ with respect to $x$ on the closed interval $[ 1,4 ]$ is
   (A) $- \frac { 1 } { 4 }$
   (B) $\frac { 1 } { 2 } \ln 2$
   (C) $\frac { 2 } { 3 } \ln 2$
   (D) $\frac { 2 } { 5 }$
   (E) 2
4. Let $R$ be the region in the first quadrant enclosed by the $x$-axis and the graph of $y = \ln \left( 1 + 2 x - x ^ { 2 } \right)$. If Simpson's Rule with 2 subintervals is used to approximate the area of $R$, the approximation is
   (A) 0.462
   (B) 0.693
   (C) 0.924
   (D) 0.986
   (E) 1.850
5. Let $f ( x ) = \int _ { - 2 } ^ { x ^ { 2 } - 3 x } e ^ { t ^ { 2 } } d t$. At what value of $x$ is $f ( x )$ a minimum?
   (A) For no value of $x$
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { 3 } { 2 }$
   (D) 2
   (E) 3
6. $\lim _ { x \rightarrow 0 } ( 1 + 2 x ) ^ { \csc x } =$
   (A) 0
   (B) 1
   (C) 2
   (D) $e$
   (E) $e ^ { 2 }$

1993 AP Calculus BC: Section I

1. The coefficient of $x ^ { 6 }$ in the Taylor series expansion about $x = 0$ for $f ( x ) = \sin \left( x ^ { 2 } \right)$ is
   (A) $- \frac { 1 } { 6 }$
   (B) 0
   (C) $\frac { 1 } { 120 }$
   (D) $\frac { 1 } { 6 }$
   (E) 1
2. If $f$ is continuous on the interval $[ a , b ]$, then there exists $c$ such that $a < c < b$ and $\int _ { a } ^ { b } f ( x ) d x =$
   (A) $\frac { f ( c ) } { b - a }$
   (B) $\frac { f ( b ) - f ( a ) } { b - a }$
   (C) $f ( b ) - f ( a )$
   (D) $f ^ { \prime } ( c ) ( b - a )$
   (E) $f ( c ) ( b - a )$
3. If $f ( x ) = \sum _ { k = 1 } ^ { \infty } \left( \sin ^ { 2 } x \right) ^ { k }$, then $f ( 1 )$ is
   (A) 0.369
   (B) 0.585
   (C) 2.400
   (D) 2.426
   (E) 3.426

14. A particle moves along the $x$-axis so that its position at time $t$ is given by $x ( t ) = t ^ { 2 } - 6 t + 5$. For what value of $t$ is the velocity of the particle zero?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

24. The maximum acceleration attained on the interval $0 \leq t \leq 3$ by the particle whose velocity is given by $v ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 12 t + 4$ is
(A) 9
(B) 12
(C) 14
(D) 21
(E) 40

a) $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$
using similar triangles:
$\vec { V } = \frac { 1 } { 3 } \pi \left( \frac { 1 } { 3 } h \right) ^ { 2 } h = \frac { 1 } { 27 } \pi h ^ { 3 }$
$$\frac { d h } { d t } = h - 12 , V = \frac { 1 } { 3 } \pi r ^ { 2 } h$$
b) we want dv when $h = 3$
$$\begin{aligned} V & = \frac { 1 } { 27 } \pi h ^ { 3 } d t \\ d y & = \frac { 1 } { a } \pi h ^ { 2 } d h \\ d t & = \frac { 1 } { a t } \pi h ^ { 2 } ( h - 12 ) \\ & = \frac { 1 } { 9 } \pi \left( 3 ^ { 2 } \right) ( - 9 ) \\ & = - 9 \pi \end{aligned}$$
$- 9 \pi \mathrm { ft } ^ { 3 } / \min$
c) we want $\frac { d y } { d t }$ when $h = 3$
$$\begin{aligned} & \pi R ^ { 2 } = 400 \pi \\ & R ^ { 2 } = 400 \\ & R = 20 \end{aligned}$$
volume of culinder $= \pi R ^ { 2 } 4$
$\frac { d V } { d t } = 2 \pi R \varphi \frac { d R } { d t } + \pi R ^ { 2 } \frac { d y } { d t }$
$d V = \pi R ^ { 2 } d y \quad d R = 0$ since $R$ is a constant
$d t d t d t$
$9 \pi = 400 \pi \frac { d \| } { d t }$
$d u = q \quad t +$ Imin
d1 400
or:
$$\begin{aligned} & y = 400 \pi y \\ & \frac { d y } { d t } = 400 \pi d y \\ & \frac { d y } { d t } = \frac { 9 \pi } { 400 \pi } = \frac { 9 } { 400 } \end{aligned} \text { Himin }$$

The number of gallons, $P(t)$, of a pollutant in a lake changes at the rate $P^{\prime}(t) = 1 - 3e^{-0.2\sqrt{t}}$ gallons per day, where $t$ is measured in days. There are 50 gallons of the pollutant in the lake at time $t = 0$. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time $t = 9$? Why or why not?
(b) For what value of $t$ will the number of gallons of pollutant be at its minimum? Justify your answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.
(d) An investigator uses the tangent line approximation to $P(t)$ at $t = 0$ as a model for the amount of pollutant in the lake. At what time $t$ does this model predict that the lake becomes safe?

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 wind chill is the temperature, in degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ), a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity $v$, in miles per hour (mph). If the air temperature is $32 ^ { \circ } \mathrm { F }$, then the wind chill is given by $W ( v ) = 55.6 - 22.1 v ^ { 0.16 }$ and is valid for $5 \leq v \leq 60$. (a) Find $W ^ { \prime } ( 20 )$. Using correct units, explain the meaning of $W ^ { \prime } ( 20 )$ in terms of the wind chill. (b) Find the average rate of change of $W$ over the interval $5 \leq v \leq 60$. Find the value of $v$ at which the instantaneous rate of change of $W$ is equal to the average rate of change of $W$ over the interval $5 \leq v \leq 60$. (c) Over the time interval $0 \leq t \leq 4$ hours, the air temperature is a constant $32 ^ { \circ } \mathrm { F }$. At time $t = 0$, the wind velocity is $v = 20 \mathrm { mph }$. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at $t = 3$ hours? Indicate units of measure.

For $t \geq 0$, the position of a particle moving along the $x$-axis is given by $x ( t ) = \sin t - \cos t$. What is the acceleration of the particle at the point where the velocity is first equal to 0 ?
(A) $- \sqrt { 2 }$
(B) $-1$
(C) 0
(D) 1
(E) $\sqrt { 2 }$

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.

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 $t = 8$?

The height of a tree at time $t$ is given by a twice-differentiable function $H$, where $H ( t )$ is measured in meters and $t$ is measured in years. Selected values of $H ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(years)

& 2 & 3 & 5 & 7 & 10 \hline

$H ( t )$

(meters)

& 1.5 & 2 & 6 & 11 & 15 \hline \end{tabular}
(a) Use the data in the table to estimate $H ^ { \prime } ( 6 )$. Using correct units, interpret the meaning of $H ^ { \prime } ( 6 )$ in the context of the problem.
(b) Explain why there must be at least one time $t$, for $2 < t < 10$, such that $H ^ { \prime } ( t ) = 2$.
(c) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the average height of the tree over the time interval $2 \leq t \leq 10$.
(d) The height of the tree, in meters, can also be modeled by the function $G$, given by $G ( x ) = \frac { 100 x } { 1 + x }$, where $x$ is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?

A cylindrical barrel with a diameter of 2 feet contains collected rainwater. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modeled by $\dfrac{dh}{dt} = -\dfrac{1}{10}\sqrt{h}$, where $h$ is measured in feet and $t$ is measured in seconds. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.)
(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.
(b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.
(c) At time $t = 0$ seconds, the height of the water is 5 feet. Use separation of variables to find an expression for $h$ in terms of $t$.

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$, where $r(t)$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r'(t)$, the rate of change of the radius, over the time interval $0 \leq t \leq 12$.

$t$ (days)	0	3	7	10	12
$r'(t)$ (centimeters per day)	$-6.1$	$-5.0$	$-4.4$	$-3.8$	$-3.5$

(a) Approximate $r''(8.5)$ using the average rate of change of $r'$ over the interval $7 \leq t \leq 10$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t$, $0 \leq t \leq 3$, for which $r'(t) = -6$? Justify your answer.
(c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of $\int_{0}^{12} r'(t)\,dt$.
(d) The height of the cone decreases at a rate of 2 centimeters per day. At time $t = 3$ days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t = 3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac{1}{3}\pi r^2 h$.)

Stephen swims back and forth along a straight path in a 50-meter-long pool for 90 seconds. Stephen's velocity is modeled by $v(t) = 2.38e^{-0.02t}\sin\left(\frac{\pi}{56}t\right)$, where $t$ is measured in seconds and $v(t)$ is measured in meters per second.
(a) Find all times $t$ in the interval $0 < t < 90$ at which Stephen changes direction. Give a reason for your answer.
(b) Find Stephen's acceleration at time $t = 60$ seconds. Show the setup for your calculations, and indicate units of measure. Is Stephen speeding up or slowing down at time $t = 60$ seconds? Give a reason for your answer.
(c) Find the distance between Stephen's position at time $t = 20$ seconds and his position at time $t = 80$ seconds. Show the setup for your calculations.
(d) Find the total distance Stephen swims over the time interval $0 \leq t \leq 90$ seconds. Show the setup for your calculations.

A particle moves along the $x$-axis so that its velocity at time $t \geq 0$ is given by $v(t) = \ln\left(t^2 - 4t + 5\right) - 0.2t$.
(a) There is one time, $t = t_R$, in the interval $0 < t < 2$ when the particle is at rest (not moving). Find $t_R$. For $0 < t < t_R$, is the particle moving to the right or to the left? Give a reason for your answer.
(b) Find the acceleration of the particle at time $t = 1.5$. Show the setup for your calculations. Is the speed of the particle increasing or decreasing at time $t = 1.5$? Explain your reasoning.
(c) The position of the particle at time $t$ is $x(t)$, and its position at time $t = 1$ is $x(1) = -3$. Find the position of the particle at time $t = 4$. Show the setup for your calculations.
(d) Find the total distance traveled by the particle over the interval $1 \leq t \leq 4$. Show the setup for your calculations.

The wind chill is the temperature, in degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ), a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity $v$, in miles per hour (mph). If the air temperature is $32 ^ { \circ } \mathrm { F }$, then the wind chill is given by $W ( v ) = 55.6 - 22.1 v ^ { 0.16 }$ and is valid for $5 \leq v \leq 60$. (a) Find $W ^ { \prime } ( 20 )$. Using correct units, explain the meaning of $W ^ { \prime } ( 20 )$ in terms of the wind chill. (b) Find the average rate of change of $W$ over the interval $5 \leq v \leq 60$. Find the value of $v$ at which the instantaneous rate of change of $W$ is equal to the average rate of change of $W$ over the interval $5 \leq v \leq 60$. (c) Over the time interval $0 \leq t \leq 4$ hours, the air temperature is a constant $32 ^ { \circ } \mathrm { F }$. At time $t = 0$, the wind velocity is $v = 20 \mathrm { mph }$. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at $t = 3$ hours? Indicate units of measure.

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function $r$ of time $t$, where $t$ is measured in minutes. For $0 < t < 12$, the graph of $r$ is concave down. The table below gives selected values of the rate of change, $r'(t)$, of the radius of the balloon over the time interval $0 \leq t \leq 12$. The radius of the balloon is 30 feet when $t = 5$.

$t$ (minutes)	0	2	5	7	11	12
$r'(t)$ (feet per minute)	5.7	4.0	2.0	1.2	0.6	0.5

(Note: The volume of a sphere of radius $r$ is given by $V = \frac{4}{3}\pi r^3$.)
(a) Estimate the radius of the balloon when $t = 5.4$ using the tangent line approximation at $t = 5$. Is your estimate greater than or less than the true value? Give a reason for your answer.
(b) Find the rate of change of the volume of the balloon with respect to time when $t = 5$. Indicate units of measure.
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate $\int_{0}^{12} r'(t)\, dt$. Using correct units, explain the meaning of $\int_{0}^{12} r'(t)\, dt$ in terms of the radius of the balloon.
(d) Is your approximation in part (c) greater than or less than $\int_{0}^{12} r'(t)\, dt$? Give a reason for 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).

4. A squirrel starts at building $A$ at time $t = 0$ and travels along a straight, horizontal wire connected to building $B$. For $0 \leq t \leq 18$, the squirrel's velocity is modeled by the piecewise-linear function defined by the graph above.
(a) At what times in the interval $0 < t < 18$, if any, does the squirrel change direction? Give a reason for your answer.
(b) At what time in the interval $0 \leq t \leq 18$ is the squirrel farthest from building $A$ ? How far from building $A$ is the squirrel at that time?
(c) Find the total distance the squirrel travels during the time interval $0 \leq t \leq 18$.
(d) Write expressions for the squirrel's acceleration $a ( t )$, velocity $v ( t )$, and distance $x ( t )$ from building $A$ that are valid for the time interval $7 < t < 10$.

WRITE ALL WORK IN THE EXAM BOOKLET.

© 2010 The College Board. Visit the College Board on the Web: \href{http://www.collegeboard.com}{www.collegeboard.com}.

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W(t)$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55^{\circ}\mathrm{F}$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W(t)$ at selected times $t$ for the first 20 minutes are given in the table below.

$t$ (minutes)	0	4	9	15	20
$W(t)$ (degrees Fahrenheit)	55.0	57.1	61.8	67.9	71.0

(a) Use the data in the table to estimate $W'(12)$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int_{0}^{20} W'(t)\, dt$. Using correct units, interpret the meaning of $\int_{0}^{20} W'(t)\, dt$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac{1}{20} \int_{0}^{20} W(t)\, dt$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac{1}{20} \int_{0}^{20} W(t)\, dt$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W'(t) = 0.4\sqrt{t}\cos(0.06t)$. Based on the model, what is the temperature of the water at time $t = 25$?

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W ( t )$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55 ^ { \circ } \mathrm { F }$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W ( t )$ at selected times $t$ for the first 20 minutes are given in the table above.

$t$ (minutes)	0	4	9	15	20
$W ( t )$ (degrees Fahrenheit)	55.0	57.1	61.8	67.9	71.0

(a) Use the data in the table to estimate $W ^ { \prime } ( 12 )$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$. Using correct units, interpret the meaning of $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W ^ { \prime } ( t ) = 0.4 \sqrt { t } \cos ( 0.06 t )$. Based on the model, what is the temperature of the water at time $t = 25$ ?

In a cardboard disk of radius $R$, we cut out an angular sector corresponding to an angle of measure $\alpha$ radians. We overlap the edges to create a cone of revolution. We wish to choose the angle $\alpha$ to obtain a cone of maximum volume.
We call $\ell$ the radius of the circular base of this cone and $h$ its height. We recall that:

• the volume of a cone of revolution with base a disk of area $\mathscr{A}$ and height $h$ is $\frac{1}{3}\mathscr{A}h$.
• the length of an arc of a circle of radius $r$ and angle $\theta$, expressed in radians, is $r\theta$.

1. We choose $R = 20\mathrm{~cm}$. a. Show that the volume of the cone, as a function of its height $h$, is $$V(h) = \frac{1}{3}\pi\left(400 - h^2\right)h.$$ b. Justify that there exists a value of $h$ that makes the volume of the cone maximum. Give this value. c. How should we cut the cardboard disk to have maximum volume? Give an approximation of $\alpha$ to the nearest degree.
2. Does the angle $\alpha$ depend on the radius $R$ of the cardboard disk?

The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ In this question, round both results to the nearest thousandth. a. Calculate $f(20)$. b. Determine the maximum rate of $\mathrm{CO}_2$ present in the room during the experiment.

Exercise 2

During a laboratory experiment, a projectile is launched into a fluid medium. The objective is to determine for which firing angle $\theta$ with respect to the horizontal the height of the projectile does not exceed 1.6 meters. Since the projectile does not move through air but through a fluid, the usual parabolic model is not adopted. Here we model the projectile as a point that moves, in a vertical plane, on the curve representing the function $f$ defined on the interval $[0; 1[$ by: $$f(x) = bx + 2\ln(1-x)$$ where $b$ is a real parameter greater than or equal to 2, $x$ is the abscissa of the projectile, $f(x)$ its ordinate, both expressed in meters.

1. The function $f$ is differentiable on the interval $[0; 1[$. We denote $f'$ its derivative function.
   We admit that the function $f$ has a maximum on the interval $[0; 1[$ and that, for every real $x$ in the interval $[0; 1[$: $$f'(x) = \frac{-bx + b - 2}{1 - x}$$ Show that the maximum of the function $f$ is equal to $b - 2 + 2\ln\left(\frac{2}{b}\right)$.
2. Determine for which values of the parameter $b$ the maximum height of the projectile does not exceed 1.6 meters.
3. In this question, we choose $b = 5.69$.
   The firing angle $\theta$ corresponds to the angle between the abscissa axis and the tangent to the curve of the function $f$ at the point with abscissa 0. Determine an approximate value of the angle $\theta$ to the nearest tenth of a degree.