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**Subject of the mathematics patent**

**Session 2017**

Foreign centers

**Exercise 1: (6 points)**

For each of the following statements, say whether it is true or false. Each answer must be justified.

**Assertion 1:**

A carpenter takes the following measurements in the corner of a wall 1 meter above the floor to

build a shelf ABC :

AB = 65 cm; AC = 72 cm and BC = 97 cm

He thinks for a few minutes and ensures that the shelf has a right angle.

**Assertion 2:**

Building standards require that the slope of a roof, represented here by

the angle must be between 30° and 35°.

A cross-section of the roof is shown opposite:

AC = 6 m and AH = 5 m.

H is the middle of [AB].

The carpenter claims that his construction meets the standard.

**Assertion 3:**

A painter wishes to repaint the shutters of a house. He finds that he uses of the can to put a coat of paint on the inside and outside of a shutter.

He has to paint his 4 pairs of shutters and put on each shutter 3 layers of paint.

He says he needs 2 cans of paint.

**Exercise 2: (7 points)**

**Part 1:**

To carry out a study on different insulators, a company makes 3 models of

identical with the exception of the insulation which differs in each house.

model. These 3 models are then placed in a cold room regulated at 6°C.

A temperature reading is taken, which allows us to construct the 3 graphs

following :

1. What was the temperature of the models before they were put in the cold room?

2. Did this experience last more than 2 days? Justify your answer.

3. Which model contains the best performing insulation? Justify your answer.

**Part 2:**

To comply with the RT2012 standard for low-energy houses (Bâtiments Basse

Consumption), it is necessary that the thermal resistance of the walls noted is higher

or equal to 4. To calculate this thermal resistance, we use the relation :

where is the thickness of the insulation in meters and c is the conductivity coefficient

thermal insulation. This coefficient allows to know the performance of the insulation.

1. Noa has chosen glass wool as insulation, whose conductivity coefficient

thermal is: = 0.035. He wants to put 15 cm of glass wool on his walls.

Does his house meet the RT2012 standard for low-energy houses?

2. Camille wishes to obtain a thermal resistance of 5 ( = 5). She has chosen as

cork insulation with a thermal conductivity coefficient of: = 0.04.

What thickness of insulation should she put on her walls?

**Exercise 3: (6 points)**

Here are the dimensions of four solids:

- A pyramid of 6 cm height whose base is a rectangle of 6 cm length and 3 cm width.
- A cylinder of 2 cm radius and 3 cm height.
- A cone of 3 cm radius and 3 cm height.
- A ball with a radius of 2 cm.

1. a) Represent approximately the first three solids

like the example on the right.

b) Place the given dimensions on the representations.

2. Rank these four solids in order of increasing volume.

**Exercise 4: (4 points)**

A manufacturer of electric roller shutters carries out a statistical study to know

their reliability. It therefore runs a sample of 500 flaps without stopping, until

a possible breakdown. He enters the results in the spreadsheet below:

1. What formula should be entered in cell H2 of the spreadsheet to obtain the total number of flaps tested?

2. An employee randomly picks a pane from this sample.

What is the probability that this shutter will run more than 3000 up and downs?

3. The manufacturer considers its dampers reliable if more than 95% of the dampers operate for more than

1000 ascents and descents. Is this set of roller shutters reliable? Explain your reasoning.

**Exercise 5: (6 points)**

Sarah has just built a swimming pool in the shape of a straight block, 8 m long, 4 m wide and 1.80 m deep. She now wants to fill her pool.

So she installs her garden hose there.

Sarah noticed that with her garden hose, she can fill a 10 liter bucket in 18 seconds.

To fill the pool, a space of 20 cm must be left between the surface of the water and the top of the pool.

Does it take more or less than a day to fill the pool? Justify your answer.

**Exercise 6: (9 points)**

1. Verify that d is approximately equal to 71 to the nearest unit.

2. A point in a window of execution of your program has its abscissa which can vary from – 240 to 240 and its ordinate which can vary from -180 to 180.

What is the largest integer that can be used in the main program to make the “street” plot fit in the window on your computer where the program is running?

You can draw on your copy all the diagrams (freehand or not) which have

to answer the previous question and add all useful information

(values, encodings, additional lines, point names…)

3. Attention, this question is independent of the previous questions and the “house” is slightly different.

If you want to add a chimney outlet to the layout of the house to make it more realistic, you must do a minimum of calculations to avoid surprises.

Examples:

It is assumed that:

– the points H, E and A are aligned;

– the points C, M and A are aligned;

– [CH] and [EM] are perpendicular to [HA];

– AM = 16 ;

– MC = 10 ;

– = 30°.

Calculate EM, HC and HE in order to obtain a nice chimney exit.

**Exercise 7: (7 points)**

Bob has to redo the tiles in his kitchen, the shape of which is a rectangle of 4 m

by 5 m.

He chose his tiles in a store. The salesman tells him to order

5% more tile to compensate for losses due to cutting.

The selected tiles are sold in packages that cover 1.12 m² and

each package costs 31 €.

1. Show that Bob must order at least 21 m² of tiles.

2. How many packages of tiles should he buy?

3. How much will it cost to buy tiles for his kitchen?

4. Bob then goes to another store to buy the rest of his materials.

Complete the invoice below:

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