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**A subject of the patent of maths 2017 in order to revise and prepare its patent of colleges free by revising the functions, the literal calculation in 3rd.As well as the volumes of the solids of space in class of third.
**

**Exercise 1 (4 points)**

The following two calculation programs are proposed:

1. Show that if we choose 3 as the starting number, both programs give 25 as the result.

2. With program A, what number must be chosen at the beginning so that the result is 0?

3. Germaine claims that, for any starting number, these two programs give the

same result. Is she right? Justify.

**Exercise 2 (3 points)**

A guitar string is subjected to a tension T, expressed in Newton (N), which allows to obtain a sound when the string is plucked. This sound, more or less acute, is characterized by its frequency f expressed in Hertz (Hz).

The function which, to a voltage T, associates its frequency f is defined by the relation: .

The graphical representation of this function is given below.

1. Determine graphically an approximate value of the tension to be applied on the string to obtain an “A3”.

2. Determine by calculation the score obtained if the string is pinched with a tension of about 220 N.

3. The string breaks if the tension is higher than 900 N. What is the maximum frequency it can emit before it breaks?

**Exercise 3 (6 points)**

Germaine wants to build a staircase to go upstairs in her apartment.

To do this, it needs to know the dimensions of the silt (the board in which the silt will be placed).

fix the steps of this staircase).

She makes the sketch below.

On this sketch,

– The silt is represented by the ACDE quadrilateral.

– The lines (ED) and (AC) are parallel.

– Points E, A and B are aligned as well as points D, C and B.

1. Prove that ED = 450 cm

2. Calculate the two dimensions AC and AE of this board. Round the results to the nearest centimeter.

**Exercise 4 (3 points)**

A silicone mold has 6 cavities. All cavities are

identical.

Each cavity has the shape of a half-sphere with a radius of 4 cm.

**Reminders:**

Volume of a ball of radius r:.

1. Show that the volume of a cavity is about 134 .

2. Germaine has prepared 1 L of dough. She wants to fill each cavity of the mold to of its volume.

Does it have enough batter for all 6 cavities in the mold? Justify the answer.

**Exercise 5 (5 points)**

Let be the functions f, g and h defined by :

.

Using a spreadsheet program, Marc constructed a table of values for these functions. He stretched the formulas he had entered in cells B2, B3 and B4 to the right.

1. Use the spreadsheet to determine the value of ℎ(-2).

2. Write the calculations showing that g(-3) = 47.

3. Make a sentence with the word “antecedent” or the word “image” to translate the equality g(-3) = 47.

4. What formula did Mark enter in cell B4?

5. Deduce from the table above a solution to the equation 3x²-9x-7=5x-7.

**Exercise 6 (4 points)**

Mathilde bought a pair of pants on January 6th during the first week of the sales. She got a 30% discount on the original price. She paid 60,20 €.

1. How much did the pants cost in December, before the sales?

2. The following week, the same store lowers its prices by an additional 10% (compared to the sale price). Annie will buy the same pants as Mathilde. How much will she pay?

3. Is it true that Annie received a 40% discount from the original price? Justify the answer.

**Exercise 7 (4 points)**

Mr. Dupont wishes to tile a rectangular terrace, length 715 cm and width 520 cm. Not being very patient, he wants to use the largest possible size of tile for this purpose so that he can spend as little time as possible laying them. Also, he would like to avoid having to cut these tiles. A search on various sites allows him to ensure that he can buy square tiles of any possible size (in whole numbers of centimeters).

1. Mr. Smith talks to his neighbor, Mr. Smith, about his project. The latter advises him to choose tiles of 65 cm on a side. Justify this advice, write down all the calculations.

2. How many tiles will Mr. Dupont need to tile his entire terrace?

3. The store where Mr. Smith and Mr. Smith (who comes to help his neighbor) sell the tiles in sets of 7, each set costing 16.90 €. What price will Mr. Dupont pay?

**Exercise 8 (7 points)**

Laurent sets up as a goat farmer to produce milk to make cheese.

**Part 1: Milk production**

Document 1

Alpine goat :

Milk production: 1.8 liters of milk per day per goat on average

Grazing: 12 goats maximum per hectare

Document 2

Simplified plan of grazing areas

Document 3

1 hectare (1 ha) = 10,000 m².

1. Prove that Laurent can own at most 247 goats.

2. Under these conditions, how many liters of milk

can expect to produce per day on average?

**Part 2 : milk storage**

Laurent wants to buy a cylindrical tank to store the milk of his goats. He can choose between 2 models:

– tank A : capacity 585 L

– tank B: diameter 100 cm, height 76 cm.

Cylinder volume formula: and .

He chooses the tank with the largest capacity, which one will he choose?

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