# Brevet Amérique 2016 : subject of the mathematics patent

SOUTH AMERICA
Mathematics certificate

December 1, 2016
Duration: 2 hours

EXERCISE 1 : 6 points
This exercise is a multiple choice questionnaire (MCQ).
For each line in the table, three answers are proposed, but only one is correct.
Any correct answer is worth 2 points.
Any inaccurate answer or lack of answer does not take away any points.
Indicate on the copy the number of the question and, without justifying, copy the correct answer.

EXERCISE 2 : 6 points
An airliner carrying passengers lands at Galeao International Airport in Rio de Janeiro.
We study the braking distance of the aircraft as a function of its speed at the time of landing.
The pilot can decide to use “fast” braking if he wants to shorten the braking distance, or “comfort” braking
more moderate and therefore more comfortable for the passengers.
The following curves give the braking distance of an aircraft as a function of its speed at the time of landing according to
the selected braking mode (comfort or fast).

1. Give by graphical reading, without justification:
(a) An approximate value for the “comfort” braking distance of the aircraft if the aircraft arrives at a speed of
.
(b) An approximate value for the landing speed of an aircraft with a “fast” braking distance of 1500 m.
2. To return to the passenger disembarkation area, the aircraft must use one of the four exits specified in
the documents below:

(a) The plane lands at . The pilot decides on “comfort” braking.

With the corresponding braking distance, which exit(s) will he pass?
(b) Since only output 1 is available, the pilot is considering “fast” braking. Determine with the precision of the
graph, the maximum speed with which it can land to be able to use this exit.

EXERCISE 3: 5 points
Carole wants to make a mosaic on a wall of her house. The surface to be paved is a rectangle of dimensions 108 cm
and 225 cm and must be entirely covered by square tiles of the same size without cutting.
1. Can Carol use 3 cm squares? 6 cm on a side?
2. What is the maximum size of tile that Carol can lay? How many tiles will she use?

EXERCISE 4 : 3 points
Cristo Redentor, a Brazilian symbol, is a large statue overlooking the city of Rio that stands atop Mount Corcovado.
At the foot of the monument, Julien and Magali want to measure the height of the statue (including the base). Julien who measures 1.90 m,
stands a few meters in front of the statue. Magali places her gaze at ground level so that she can see the
Cristo summit (S) and that of Julien’s head (T) aligned; it is then located at 10 m from the statue and 50 cm from Julian. The
situation is modeled below by the figure which is not to scale.

Determine the height SC of the statue assuming that the monument and Julian are perpendicular to the ground.

EXERCISE 5 : 6 points
To climb to the top of Corcovado and access the statue from the center of Rio, you can take a minibus. The price
A ticket in Brazilian Real (R\$), Brazilian currency, includes transportation to the site and access to the monument.
The following documents are given.

1. Determine the price of the visit for an adult on 09/02/2016.
2. Determine the price of the visit for a child between the ages of 6 and 11 on 09/02/2016.

EXERCISE 6 : 4 points
Inaugurated in 1950, the Maracanà stadium is a mythical place, place of great sports events such as the World Cup
2014 or the 2016 Olympic Games. It is a structure of oval form of dimensions 317 m and 279 m for a height
of 32 m with a floor area of approximately 69,500 m².

On the famous beach of Copacabana, in Rio, you can admire many sand sculptures. One of the sculptors wants to make a 1/300 scale reproduction of the stadium.
1. What will be the dimensions, rounded to the centimeter, of this reproduction?
(a) What will be the area? We will give the result in m², rounded to the hundredth.
(b) The sculptor has a space of 1 m².

Is it certain that it can be reproduced? The answer will be briefly justified.

EXERCISE 7: 7 points
The Sugarloaf Mountain is a peak located in Rio on the side of the sea. It culminates at 396 meters of altitude and is accessible by a
cable car composed of two sections.

The second section of the cable car from point U to the summit S of the peak is shown above.
We give: Altitude of point S: 396 m US = 762 m
Altitude of the point U : 220 m The triangle UOS is rectangular in O.

1. Determine the angle DUS that the cable car cable forms with the horizontal. The result will be rounded to the degree.
2. Knowing that the travel time between stations U and S is 6 min 30 s, calculate the average speed of the cable car between
these two stations in meters per second. The result will be rounded to the meter per second.
3. The attendance at the Sugarloaf was recorded for one day and this information was entered into a spreadsheet of a

We have entered in cell H2 the formula: =SUM(B2:G2)
3.a Interpret the number calculated with this formula.
3.b What is the number of visitors between 12:00 and 14:00?
3.c A formula must be entered to calculate the average number of visitors per hour on that day. Among the proposals
Copy the appropriate one without justification:
MEAN(B2:G2) =MEAN(B2:G2)
MEAN(B2:G2)/2 =MEAN(B2:G2)/2

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