Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time $t$, $0 \leq t \leq 6$, is given by a differentiable function $C$, where $t$ is measured in minutes. Selected values of $C ( t )$, measured in ounces, are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 1 & 2 & 3 & 4 & 5 & 6 \hline

$C ( t )$

(ounces)

& 0 & 5.3 & 8.8 & 11.2 & 12.8 & 13.8 & 14.5 \hline \end{tabular}
(a) Use the data in the table to approximate $C ^ { \prime } ( 3.5 )$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t$, $2 \leq t \leq 4$, at which $C ^ { \prime } ( t ) = 2$? Justify your answer.
(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$. Using correct units, explain the meaning of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$ in the context of the problem.
(d) The amount of coffee in the cup, in ounces, is modeled by $B ( t ) = 16 - 16 e ^ { - 0.4 t }$. Using this model, find the rate at which the amount of coffee in the cup is changing when $t = 5$.

Concert tickets went on sale at noon $( t = 0 )$ and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time $t$ is modeled by a twice-differentiable function $L$ for $0 \leq t \leq 9$. Values of $L ( t )$ at various times $t$ are shown in the table below.

$t$ (hours)	0	1	3	4	7	8	9
$L ( t )$ (people)	120	156	176	126	150	80	0

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. $( t = 5.5 )$. Show the computations that lead to your answer. Indicate units of measure.
(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.
(c) For $0 \leq t \leq 9$, what is the fewest number of times at which $L ^ { \prime } ( t )$ must equal 0 ? Give a reason for your answer.
(d) The rate at which tickets were sold for $0 \leq t \leq 9$ is modeled by $r ( t ) = 550 t e ^ { - t / 2 }$ tickets per hour. Based on the model, how many tickets were sold by 3 P.M. ( $t = 3$ ), to the nearest whole number?

Exercise 3 (3 points)

Part A:

A health control agency is interested in the number of bacteria of a certain type contained in fresh cream. It performs analyses on 10000 samples of 1 ml of fresh cream from the entire French production. The results are given in the table below:

\begin{tabular}{ l } Number of bacteria

(in thousands)

& $[100;120[$ & $[120;130[$ & $[130;140[$ & $[140;150[$ & $[150;160[$ & $[160;180[$ \hline Number of samples & 1597 & 1284 & 2255 & 1808 & 1345 & 1711 \hline \end{tabular}
Using a calculator, give an estimate of the mean and standard deviation of the number of bacteria per sample.

Part B:

The agency then decides to model the number of bacteria studied (in thousands per ml) present in fresh cream by a random variable $X$ following the normal distribution with parameters $\mu = 140$ and $\sigma = 19$.

1. a. Is this choice of modelling relevant? Argue. b. We denote $p = P(X \geqslant 160)$. Determine the value of $p$ rounded to $10^{-3}$.
   
2. During the inspection of a dairy, the health control agency analyzes a sample of 50 samples of 1 ml of fresh cream from the production of this dairy; 13 samples contain more than 160 thousand bacteria. a. The agency declares that there is an anomaly in the production and that it can affirm it with a probability of 0.05 of being wrong. Justify its declaration. b. Could it have affirmed it with a probability of 0.01 of being wrong?

1. Between 1998 and 2020, in France 18221965 deliveries were recorded, of which 293898 resulted in the birth of twins and 4921 resulted in the birth of at least three children. a. With a precision of $0.1\%$ calculate, among all recorded deliveries, the percentage of deliveries resulting in the birth of twins over the period 1998-2020. b. Verify that the percentage of deliveries that resulted in the birth of at least three children is less than $0.1\%$.

We then consider that this percentage is negligible. We call an ordinary delivery a delivery resulting in the birth of a single child. We call a double delivery a delivery resulting in the birth of exactly two children. We consider in the rest of the exercise that a delivery is either ordinary or double. The probability of an ordinary delivery is equal to 0.984 and that of a double delivery is then equal to 0.016. The probabilities calculated in the rest will be rounded to the nearest thousandth.
2. We admit that on a given day in a maternity ward, $n$ deliveries are performed. We consider that these $n$ deliveries are independent of each other. We denote $X$ the random variable that gives the number of double deliveries performed that day. a. In the case where $n = 20$, specify the probability distribution followed by the random variable $X$ and calculate the probability that exactly one double delivery is performed. b. By the method of your choice that you will explain, determine the smallest value of $n$ such that $P ( X \geqslant 1 ) \geqslant 0.99$. Interpret the result in the context of the exercise.
3. In this maternity ward, among double births, it is estimated that there are $30\%$ monozygotic twins (called ``identical twins'' which are necessarily of the same sex: two boys or two girls) and therefore $70\%$ dizygotic twins (called ``fraternal twins'', which can be of different sexes: two boys, two girls or one boy and one girl). In the case of double births, we admit that, as for ordinary births, the probability of being a girl at birth is equal to 0.49 and that of being a boy at birth is equal to 0.51. In the case of a double birth of dizygotic twins, we also admit that the sex of the second newborn of the twins is independent of the sex of the first newborn. We randomly choose a double delivery performed in this maternity ward and we consider the following events:

• $M$ : ``the twins are monozygotic'';
• $F _ { 1 }$ : ``the first newborn is a girl'';
• $F _ { 2 }$ : ``the second newborn is a girl''.

We will denote $P ( A )$ the probability of event $A$ and $\bar { A }$ the opposite event of $A$. a. Copy and complete the probability tree. b. Show that the probability that the two newborns are girls is 0.315 07. c. The two newborns are twin girls. Calculate the probability that they are monozygotic.

Exercise 1 — Part B
We randomly choose a person who came to the multisports centre on a weekend. We denote $T_1$ the random variable giving their total waiting time in minutes before access to sports activities during Saturday and $T_2$ the random variable giving their total waiting time in minutes before access to sports activities during Sunday. We admit that:

• $T_1$ follows a probability distribution with expectation $E(T_1) = 40$ and standard deviation $\sigma(T_1) = 10$;
• $T_2$ follows a probability distribution with expectation $E(T_2) = 60$ and standard deviation $\sigma(T_2) = 16$;
• the random variables $T_1$ and $T_2$ are independent.

We denote $T$ the random variable giving the total waiting time before access to sports activities over the two days, expressed in minutes. Thus we have $T = T_1 + T_2$.

1. Determine the expectation $E(T)$ of the random variable $T$. Interpret the result in the context of the exercise.
2. Show that the variance $V(T)$ of the random variable $T$ is equal to 356.
3. Using the Bienaymé-Chebyshev inequality, show that, for a person randomly chosen among those who came to the multisports centre on a weekend, the probability that their total waiting time $T$ is strictly between 60 and 140 minutes is greater than 0.77.

Question 136
A pesquisa sobre o nível de satisfação dos funcionários de uma empresa foi realizada com uma amostra de 1 000 funcionários. Os resultados obtidos estão representados no gráfico.
[Figure]
De acordo com o gráfico, o número de funcionários satisfeitos é
(A) 120 (B) 240 (C) 280 (D) 360 (E) 480

Question 139
Um estudante anotou as temperaturas máximas registradas em sua cidade durante uma semana:

Dom	Seg	Ter	Qua	Qui	Sex	Sáb
$28^\circ$	$30^\circ$	$26^\circ$	$27^\circ$	$29^\circ$	$31^\circ$	$27^\circ$

A média aritmética e a mediana das temperaturas máximas registradas nessa semana são, respectivamente,
(A) $28^\circ$ e $27^\circ$ (B) $28^\circ$ e $28^\circ$ (C) $28^\circ$ e $29^\circ$ (D) $29^\circ$ e $28^\circ$ (E) $29^\circ$ e $29^\circ$

Question 146
A tabela mostra a distribuição de frequências das notas de uma turma de 40 alunos em uma prova.

Nota	Frequência
4	5
5	8
6	10
7	9
8	5
9	3

A média aritmética das notas dessa turma é
(A) 5,5 (B) 6,0 (C) 6,3 (D) 6,5 (E) 7,0

Um estudante anotou as temperaturas máximas registradas em sua cidade durante uma semana:

Dia	Temperatura máxima ($^\circ$C)
Segunda	28
Terça	30
Quarta	26
Quinta	32
Sexta	29
Sábado	31
Domingo	27

A média aritmética das temperaturas máximas registradas nessa semana é
(A) 28,5 $^\circ$C (B) 29 $^\circ$C (C) 29,5 $^\circ$C (D) 30 $^\circ$C (E) 30,5 $^\circ$C

Em uma classe de 40 alunos, a nota média em uma prova foi 7,0. Se os 10 melhores alunos tiveram média 9,0, qual foi a média dos demais 30 alunos?
(A) 5,0 (B) 5,5 (C) 6,0 (D) 6,5 (E) 7,0

A mediana de um conjunto de dados $\{3, 7, 5, 9, 1, 6, 4\}$ é
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

The table below shows the evolution of annual gross revenue in the last three years of five microenterprises (ME) that are for sale.

ME	2009 (in thousands of reais)	2010 (in thousands of reais)	2011 (in thousands of reais)
Pins V	200	220	240
Candies W	200	230	200
Chocolates X	250	210	215
Pizzeria Y	230	230	230
Textiles Z	160	210	245

An investor wants to buy two of the companies listed in the table. To do so, he calculates the average of the annual gross revenue of the last three years (from 2009 to 2011) and chooses the two companies with the highest annual average.
The companies that this investor chooses to buy are
(A) Candies W and Pizzeria Y.
(B) Chocolates X and Textiles Z.
(C) Pizzeria Y and Pins V.
(D) Pizzeria Y and Chocolates X.
(E) Textiles Z and Pins V.

An irrigated coffee producer in Minas Gerais received a statistical consulting report, containing, among other information, the standard deviation of the productions from one harvest of the plots on his property. The plots have the same area of $30000\mathrm{~m}^{2}$ and the value obtained for the standard deviation was $90\mathrm{~kg}/\mathrm{plot}$. The producer must present information about the production and the variance of these productions in bags of 60 kg per hectare ($10000\mathrm{~m}^{2}$).
The variance of the productions of the plots expressed in (bags/hectare)$^{2}$ is
(A) 20.25.
(B) 4.50.
(C) 0.71.
(D) 0.50.
(E) 0.25.

The graph presents the behavior of formal employment created, according to CAGED, in the period from January 2010 to October 2010.
Based on the graph, the value of the integer part of the median of formal employment created in the period is
(A) 212952.
(B) 229913.
(C) 240621.
(D) 255496.
(E) 298041.

The grades of a teacher who participated in a selection process, in which the evaluation panel was composed of five members, are presented in the graph. It is known that each panel member assigned two grades to the teacher, one relating to specific knowledge in their area of work and another relating to pedagogical knowledge, and that the teacher's final grade was given by the arithmetic mean of all grades assigned by the evaluation panel.
Using a new criterion, this evaluation panel decided to discard the highest and lowest grades assigned to the teacher.
The new average, in relation to the previous average, is
(A) 0.25 point higher. (B) 1.00 point higher. (C) 1.00 point lower. (D) 1.25 points higher. (E) 2.00 points lower.

Five food companies are for sale. An entrepreneur, aiming to expand his investments, wishes to buy one of these companies. To choose which one he will buy, he analyses the profit (in millions of reais) of each one, as a function of their time (in years) of existence, deciding to buy the company that presents the highest average annual profit.
The table presents the profit (in millions of reais) accumulated over the time (in years) of existence of each company.

Company	Profit (in millions of reais)	Time (in years)
F	24	3.0
G	24	2.0
H	25	2.5
M	15	1.5
P	9	1.5

The entrepreneur decided to buy the company
(A) F. (B) G. (C) H. (D) M. (E) P.

A survey was conducted in the 200 hotels in a city, in which the values, in reais, of the daily rates for a standard double room and the number of hotels for each daily rate value were noted. The daily rate values were: $\mathrm{A} = \mathrm{R}\$ 200{,}00$; $\mathrm{B} = \mathrm{R}\$ 300{,}00$; $\mathrm{C} = \mathrm{R}\$ 400{,}00$ and $\mathrm{D} = \mathrm{R}\$ 600{,}00$. In the graph, the areas represent the numbers of hotels surveyed, in percentage, for each daily rate value.
The median value of the daily rate, in reais, for the standard double room in this city, is
(A) 300.00. (B) 345.00. (C) 350.00. (D) 375.00. (E) 400.00.

QUESTION 142
A student scored the following grades in four tests: 6.0, 7.5, 8.0, and 4.5. The arithmetic mean of the student's grades is
(A) 6.0
(B) 6.5
(C) 7.0
(D) 7.5
(E) 8.0

QUESTION 169
The median of the data set $\{3, 7, 5, 9, 1, 6, 4\}$ is
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

The graph shows the average daily oil production in Brazil, in million barrels, in the period from 2004 to 2010.
Estimates made at that time indicated that the average daily oil production in Brazil in 2012 would be 10\% higher than the average of the three last years shown in the graph.
If these estimates had been confirmed, the average daily oil production in Brazil, in million barrels, in 2012, would have been equal to
(A) 1.940.
(B) 2.134.
(C) 2.167.
(D) 2.420.
(E) 6.402.

A cable TV subscription seller had, in the first 7 months of the year, a monthly average of 84 subscriptions sold. Due to a company restructuring, all sellers were required to have, at the end of the year, a monthly average of 99 subscriptions sold. Faced with this, the seller was forced to increase his monthly sales average in the remaining 5 months of the year.
What should be the seller's monthly sales average in the next 5 months so that he can meet his company's requirement?
(A) 91
(B) 105
(C) 114
(D) 118
(E) 120

A person is competing in a selection process for a job opening in an office. In one of the stages of this process, he must type eight texts. The number of errors made by this person in each of the typed texts is given in the table.

Text	\begin{tabular}{ c } Number
of errors

\hline I & 2 \hline II & 0 \hline III & 2 \hline IV & 2 \hline V & 6 \hline VI & 3 \hline VII & 4 \hline VIII & 5 \hline \end{tabular}
In this stage of the selection process, candidates will be evaluated by the median value of the number of errors.
The median of the number of errors made by this person is equal to
(A) 2{,}0.
(B) 2{,}5.
(C) 3{,}0.
(D) 3{,}5.
(E) 4{,}0.

The evaluation of student performance in a university course is based on the weighted average of grades obtained in the disciplines by their respective number of credits, as shown in the table:

Evaluation	Average grade (M)
Excellent	$9 < M \leq 10$
Good	$7 \leq M \leq 9$
Regular	$5 \leq M < 7$
Poor	$3 \leq M < 5$
Very Poor	$M < 3$

A certain student knows that if he obtains a ``Good'' or ``Excellent'' evaluation, he will be able to enroll in the disciplines he desires. He has already taken the exams for 4 of the 5 disciplines in which he is enrolled, but has not yet taken the exam for discipline I, as shown in the table.

Disciplines	Grades	\begin{tabular}{ c } Number
of credits

\hline I & & 12 \hline II & 8.00 & 4 \hline III & 6.00 & 8 \hline IV & 5.00 & 8 \hline V & 7.50 & 10 \hline \end{tabular}
In order to achieve his objective, the minimum grade he must obtain in discipline I is
(A) 7.00.
(B) 7.38.
(C) 7.50.
(D) 8.25.
(E) 9.00.

Three students, X, Y, and Z, are enrolled in an English course. To evaluate these students, the teacher chose to give five tests. In order to pass this course, the student must have an arithmetic mean of the grades from the five tests greater than or equal to 6. The table shows the grades each student received on each test.

Student	$\mathbf{1}^{\mathbf{st}}$	$\mathbf{2}^{\mathbf{nd}}$	$\mathbf{3}^{\mathbf{rd}}$	$\mathbf{4}^{\mathbf{th}}$	$\mathbf{5}^{\mathbf{th}}$
X	5	5	5	10	6
Y	4	9	3	9	5
Z	5	5	8	5	6

Based on the data in the table and the information given, will be/will fail
(A) only student Y.
(B) only student Z.
(C) only students X and Y.
(D) only students X and Z.
(E) students X, Y, and Z.

The graph presents the unemployment rate (in \%) for the period from March 2008 to April 2009, obtained based on data observed in the metropolitan regions of Recife, Salvador, Belo Horizonte, Rio de Janeiro, São Paulo, and Porto Alegre.
The median of this unemployment rate, in the period from March 2008 to April 2009, was
(A) $8.1\%$
(B) $8.0\%$
(C) $7.9\%$
(D) $7.7\%$
(E) $7.6\%$