Grade 9 / Measurements, problems, complex tasks 19 exercises (100% corrected)

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Espace utilisé:200GoEspace libre 10cm20cm50cm 1. Use of powers of 10 E.7964 Hint: say whether the statement below is true or false, carefully justifying the answer. In computing, multiples of the byte are used as units of mea-surement : 1 ko =10 3 bytes ; 1 Mo =10 6 bytes ; 1 Go =10 9 bytes Contents of external hard disk : 1 000 photos of 900 ko each ; 65 videos of 700 Mo each. Computer capacity: 250GB Assertion: transfer of the entire contents of the external hard disk to the computer is not possible. 2. Use of a model E.896 The braking distance of a ve-hicle to a complete stop is given by the formula : D = 4 × V 2 1000 × K where : D : distance braking in m V : vehicle speed in km · h − 1 K : coefficient of road adhesion. 1 Determine the braking distance, rounded to the nearest decimeter, on a road with an adhesion coefficient of 0.25 when : a the vehicle is travelling at a speed of 110 km · h − 1 ; b the vehicle is travelling at a speed of 55 km · h − 1 ; 2 Can we say that there is proportionality between speed and braking distance? E.4333 The Ariane 5 rocket is a Euro-pean launcher that places satellites in orbit around the Earth. 1 During the first phase of rocket liftoff, the two thrusters on either side of the rocket body achieve an altitude of 70 km in 132 seconds. Calculate the average speed, expressed in m = s and rounded to the unit of the rocket during the first phase of the liftoff. Convert this result to km = h . 2 Release velocity is the speed that an object must be given to escape the pull of a planet. This velocity noted v is calculated using the following formula : v = 13.4 × 10 − 11 × M r + h où M is the mass of the planet in kg (for Earth, we have : M =6 × 10 24 kg ) , r is its radius in meters (for Earth, we have : r =6.4 × 10 6 m ) , v is then expressed in m = s . Ariane 5 releases a communications satellite at an alti-tude : h =1.9 × 10 6 m a Calculate: r + h . b What should be the speed of the rocket at this alti-tude? We will round to the nearest m = s . Write this result in scientific notation. 3. Some complex tasks E.5692 Indication : in this exercise, if the work is not completed, still leave a trace of the research. It will be taken into ac-count in the assessment. To build a breeze-block garden shed, a handyman needs 300 breeze-blocks of dimensions 50 cm × 20 cm × 10 cm each weigh-ing 10 kg . He buys the cinderblocks from a store located 10 km from his house. To transport them he rents a van from the store. Information 1: Characteristics of the van : 3 seats https://chingmath.fr chapExoCorrec/7964 sacados/7964 Espace utilisé:200GoEspace libre chapExoCorrec/896 sacados/896 chapExoCorrec/4333 sacados/4333 chapExoCorrec/5692 sacados/5692 10cm20cm50cm
Transportable volume dimensions ( L × ‘ × h ) : 2.60 m × 1.56 m × 1.84 m Load that can be transported : 1.7 ton. Tank volume : 80 liters. Diesel (consumption : 8 liters per 100 km ) Information 2: Van rental rates 1 jour 30 km maximum 1 jour 50 km maximum 1 jour 100 km maximum 1 jour 200 km maximum km supplé-mentaire 48 e 55 e 61 e 78 e 2 e Information 3: A liter of fuel costs 1.50 e . Determine the minimum cost of the cinder block set. E.6280 The document below shows the postal rates for sending a letter or package from Metropoli-tan France in the ˇ priority letter ı mode. Priority Letter service urgent sending mail For shipments to: France, Monaco, Andorra and postal sectors (army) . Additional airmail postage to overseas for items over 20 g . Universal service: Up to 2 kg . Timeframe : D +1 , indicative Dimensions: Minimum : 14 × 9 cm , maximum : L + ‘ + H =100 cm , with L< 60 cm Airborne complements: To zone OM 1 : Guyana, Guadeloupe, Martinique, Reunion, St. Pierre and Miquelon, St. Bartholomew, St. Martin, and Mayotte : 0.05 e per 10 g . To zone OM 2 : New Caledonia, French Polynesia, Wallis and Futuna, TAAF.: 0.11 e per 10 g . Example of complement: For a shipment of 32 g to Guadeloupe : 1.10 e +4 × 0.05 e =1.3 e These rates are based on the weight of the letter. Weight up to Net rates e 20 0.66 e 50 1.10 e 100 1.65 e 250 2.65 e 500 3.55 e 1 kg 4.65 e 2 kg 6.00 e 3 kg 7.00 e 1 Explain why the cost of sending a 75 g letter to Metropoli-tan France in ˇ prioritaire ı is 1.65 e . 2 Show that the cost of sending a 109 g letter to Mayotte in ˇ letter prioritaire ı is 3.20 e . Hint: in the question below, any record of even an incom-plete answer will be considered in the evaluation. 3 When it’s time to mail his mail to Wallis and Futuna, Loïc realizes that he forgot his credit card and only has 6 left; 76 e in his wallet. He intended to send a package of 272 g , in ˇpriority let-terı. Can he pay the corresponding amount? 4 The package has the following dimensions : L = 55 cm ; ‘ = 30 cm ; h = 20 cm The teller at the post office refuses him. Why? E.6304 To prepare a one-week stay in Naples, a couple living in Nantes noticed that the round-trip airfare from Nantes to Naples was much higher than that from Paris to Naples. They decided to study the cost of a round trip from Nantes to Paris to see if they should fly from Nantes or from Paris. Here is the information that the couple found : Information 1: Airline ticket prices and times. Round-trip flight from Nantes Departure from Nantes on 23 = 11 = 2014 : 06 h 35 Arrival in Naples on 23 = 11 = 2014 : 09 h 50 Departure from Naples on 30 = 11 = 2014 : 12 h 50 Arrival in Nantes on 30 = 11 = 2014 : 16 h 25 Price per person for round trip flight : 530 e Round trip flight from Paris Departure from Paris on 23 = 11 = 2014 : 11 h 55 Arrival in Naples on 23 = 11 = 2014 : 14 h 10 Departure from Naples on 30 = 11 = 2014 : 13 h 10 Arrival in Paris on 30 = 11 = 2014 : 15 h 30 Price per person for round trip flight : 350 e Passengers must be present 2 hours before takeoff to proceed with boarding. Information 2: Train prices and schedules for a passage Nantes -Paris (Airport) 23 Novembre Départ : 06 h 22 Prix : 51.00 e Duration : 03 h 16 direct Voyagez avec : TGV Travel Paris (Airport) - Nantes 30 Novembre Départ : 18 h 20 Prix : 42.00 e Duration : 03 h 19 direct Voyagez avec : TGV Information 3: Drive Average consumption : 6 liters at 100 km Péage Nantes-Paris : 35.90 e Distance home to Paris airport: 409 km Fuel : 1.30 e per liter Estimated time: 4 h 24 min Information 4: Paris airport parking Rate: 58 e for one week. 1 Explain why the difference between the prices of 2 airline tickets amounts to 360 e for this couple. 2 If the couple drives to the airport in Paris : a Determine the time before which he must leave Nantes. b Show that the fuel cost for this trip is 31.90 e . 3 Which travel arrangement is the most economical? https://chingmath.fr chapExoCorrec/6280 sacados/6280 chapExoCorrec/6304 sacados/6304
janvierfévriermarsavrilmaijuinjuilletaoûtseptembreoctobrenovembredécembre0100200300400500600Dépenses (en euros) ABCDEFGHPortedelargeur1m4m5m5m1m3m E.6314 A couple bought a house with a swimming pool in order to rent it. For this purchase, the couple took out a loan from their bank. They rent the house from June to September and the house remains unoccupied the rest of the year. Information 1: Expenses related to this house for the year 2013 The chart below shows, for each month, the total expenses due to various taxes, subscriptions (electricity, heating, wa-ter, internet) , filling and heating the pool. Information 2: Monthly loan repayment Each month, the couple must pay 700 euros to their bank to repay the loan. Information 3: Rental rate for the house Rentals are Saturday through Saturday. The couple rents their home from Saturday 7 June to Saturday 27 September 2014 . The polur rates for renting this house are as follows : Début Fin Nombre de semaines Prix de la location 07/06/2014 05/07/2014 4 semaines 750 euros per week 05/07/2014 23/08/2014 7 semaines . . . euros per week 25/08/2014 27/09/2014 5 semaines 750 euros per week For the year 2014 , with the increase in various rates and taxes, the couple anticipates that the amount of expenses related to the house will be 6 % higher than that for 2013 . 1 Explain why the total home-related expenses will be 4 505 e in 2014 . 2 It is assumed that the couple manages to rent out their house during all weeks of the rental period. At what min-imum rate (rounded up to the nearest ten euros) shouldhe rent his house between 5 = 07 and 23 = 08 to cover the costs incurred by the house over the entire year 2014 ? 4. With geometry E.9290 Mr. Chapuis wants to change the tiles and baseboards in the living room of his apartment. To do this, he needs to buy tiles, olle and wooden skirting boards that will be nailed down. He has the following docu-ments : Document 1: plan, the room corresponds to the shaded area The above diagram is not to scale. Document 2: Carrelage Size of a tile: 50 cm × 50 cm Thickness of a tile: 0.9 cm Packaging: 1.5 m 2 per boite price : 19.95 e per box. Plinthe Shape : rectangular in length 1 m Sold by the unit Price : 2.95 e the wooden baseboard. Document 3: Tile adhesive Packaging: bag of 25 kg Yield (area that can be glued) : 4 m 2 per bag Price : 22 e per bag Pack of nails for baseboards Price : 5.50 e per package Note: A skirting board is a low decorative element at-tached to the bottom of walls along the floor. 1 a Noting that the length GD is equal to 7 m , deter-mine the area of the triangle BCH . b Show that the area of the room is 32 m 2 . 2 To avoid running out of tile and glue, the vendor advises Mr. Chapuis to allow for an area 10 % larger than the area calculated in question 1 . Mr. Chapuis needs to buy whole boxes and whole bags. Determine the number of boxes of tile and the number of bags of glue to be purchased. https://chingmath.fr chapExoCorrec/6314 sacados/6314 janvierfévriermarsavrilmaijuinjuilletaoûtseptembreoctobrenovembredécembre0100200300400500600Dépenses (en euros) chapExoCorrec/9290 sacados/9290 ABCDEFGHPortedelargeur1m4m5m5m1m3m
sol du centrecommercialTrottoir roulantSol duparkingHCP25m4m C1sableC2C3S1S2 3 The vendor also recommends taking a margin of 10 % on the length of the baseboards. Determine the total number of baseboards Mr. Chapuis must purchase to go around the room. We specify that there is no baseboard on the door. 4 How much does Mr. Chapuis have to spend, knowing that he can make do with a package of nails? Round the answer to the nearest euro. E.6284 In this exercise, any trace of even unsuccessful research will be considered in the assessment. The managers of a shopping center have built an underground parking lot and want to install a moving walkway to access the shopping center from the parking lot. People using this moving sidewalk should take no more than 1 minute to access the mall. The situation is shown in the diagram below. Rolling sidewalk characteristics : Model 1 : Maximum angle of inclination with the horizontal: 12 o . Speed : 0.5 m = s Model 2 : Maximum angle of inclination with the horizontal: 6 o . Speed : 0.75 m = s Would either of these two models be appropriate to equip this shopping center? Justify. 5. With the quotient quantities E.6300 To prepare his trip to Mar-seille, Julien uses a website to choose the best itinerary. Here is the result of his search : Calculate your itiné-raire Départ 59 000 Lille France Arrivée 13 000 Marseille France 59 000 Lille - 13 000 Marseille Coût estimé : Péage 73.90 e Carburant 89.44 e Time: 8 h 47 including 8 h 31 on highway Distance : 1004 km including 993 km at autoroute 1 What average speed, rounded to the nearest km = h , does this route predict for the highway portion of the trip? 2 Knowing that road safety recommends at least one break of 10 to 20 minutes every two hours of driving, what should be the minimum amount of time that Julian should plan for his trip? 3 Hint: for this question, show on the copy the approach used. Any trace of research will be taken into account during the evaluation even if the work is not completely finished. Knowing that his car’s tank has a capacity of 60 ‘ and that a liter of gasoline costs 1.42 e , can he make the trip on one tank of gasoline based on the data on the website? E.9291 An hourglass consists of : Two cylinders C 1 and C 2 with a height of 4.2 cm and a diameter of 1.5 cm One cylinder C 3 Two hemispheres S 1 and S 2 with a diameter of 1.5 cm . Reminder : the volume V of a cylinder with a base area of B and a height of h : V = B × h Hint: we will use : ı ≈ 3.1416 1 Initially, the sand fills two-thirds of the cylinder C 2 . Show that the volume of sand is approximately 4.95 cm 3 2 The hourglass is turned over. Assuming that the flow rate of the sand is constant and equal to 1.98 cm 3 = min , calculate the time in minutes and seconds that it will take for the sand to flow into the lower cylinder. https://chingmath.fr chapExoCorrec/6284 sacados/6284 Asie Juin 2014 7 points sol du centrecommercialTrottoir roulantSol duparkingHCP25m4m chapExoCorrec/6300 sacados/6300 chapExoCorrec/9291 sacados/9291 C1sableC2C3S1S2
Schéma de la piste de cet hippodrome:Segment de longueur:850mRayon:40m ABCDEFGHIJKL5m5m E.10890 A racetrack is a place where horse races are held. We are interested in the list of a racetrack. This track consists of : two straight lines modeled by segments of 850 meters ; two turns modeled by two semicircles with a radius of 40 meters. 1 Show that the length of one lap of the track is approxi-mately 1 951 m . Hint: we will use : ı ≈ 3.1416 2 A horse runs one lap of the track in 2 min 9 s . a Calculate the average speed of this horse over one lap of the track, expressed in ( m = s ) and rounded to the nearest whole number. b Convert this speed to kilometers per hour ( km = h ) , rounded to the nearest whole number. 3 We assume that the surface area of the track is approxi-mately 73 027 m 2 . We want to sow grass over the entire surface of the track. We must choose bags of grass seed from among the three brands below : Area covered per bag Price per bag Brand A 500 m 2 141.95 e Brand B 400 m 2 87.90 e Brand C 300 m 2 66.50 e Which brand should be chosen to keep costs as low as possible? E.10807 To obtain the octagon EFGHIJKL shown opposite, four identical isosceles right triangles are removed from the corners of a square ABCD with side length 5 m . Given : ; EF = 2 ; 2 m 1 a Show that the length AE is equal to 1 ; 4 m . b Show that the area of triangle AEL is equal to 0 ; 98 m 2 . c Deduce that the area of the shaded octagon is equal to 21 ; 08 m 2 . 2 This octagon has the same dimensions as the surface area of a swimming pool with a height of 1 ; 50 m . We want to fill this pool to three-quarters of its height. a Show that the volume of water required is approxi-mately equal to 24 m 3 . b Given that the flow rate of the tap used to fill the pool is 12 L = min , calculate the time required to fill these 24 m 3 of water. Give the result in hours and minutes. Reminder : 1 m 3 = 1 000 L 6. With product sizes E.9289 Photovoltaic panels produce electricity from solar radiation. A common unit for measuring electrical energy is the kilowatt-hour, abbreviated as kWh . Most often, the electricity generated is not used directly, but sold for distribution into the collective power grid. The kWh purchase price, given in euro cents, depends on the type of installation and its total power, as well as the date of instal-lation of the photovoltaic panels. This kWh purchase price is given in the table below. Tarfis of a kWh in cents . Date d’installation Type d’ insta-llation Puissance totale du 01/01/15 au 31/03/15 du 01/04/15 au 30/06/15 du 01/07/15 au 30/09/15 du 01/10/15 au 31/12/15 Type A 0 à 9 kW 26.57 16.17 25.78 25.39 Type B 0 à 36 kW 13.46 13.95 14.7 14.4 36 à 100 kW 12.79 13.25 13.96 13.68 In May 2015, a B solar power plant is installed, with a capac-ity of 28 kW . Verify that the purchase price of 31 420 kWh is approximately 4 383 e . https://chingmath.fr chapExoCorrec/10890 sacados/10890 Schéma de la piste de cet hippodrome:Segment de longueur:850mRayon:40m chapExoCorrec/10807 sacados/10807 Martinique Juillet 2024 ABCDEFGHIJKL5m5m chapExoCorrec/9289 sacados/9289
Pays APays BPays CPays DPays E05101520253035404550 123456789ABCDTyped’énergieNucléaireThermique(gas,∏oulcharbon)HydrauliqueEolienSolaireBioénergiesTotalen2017379,153,953,524,19,29,5529,3Productionélectrique(enTWh)en2018393,239,468,327,810,29,7548,6en2019379,542,66034,111,69,9537,7 FormuleAFormuleBFormuleC0;50epourchaque kilomètreparcouruForfait ∏xe300epuis0;25epourchaque kilomètreparcouruForfait ∏xe de900epour unkilométrageillimité GrenobleLilleMarseilleNantesParisStrasbourgBordeaux6757925553385469077712807415855061005584215498909772803379864442 Information1Information2Information3Prix moyen dugazole en 20231;87epar litreVoitureproposéeType decarburant:gazole.Consommation:5;6‘pour100kmCoût totalpour les péages115;80e 7. Share E.10893 The diagram below shows so-lar photovoltaic energy production in TWh (Terawatt-hour) of the five largest producers in the European Union, which comprises twenty-eight countries in 2019 . 1 With the precision allowed by the graph, give approxi-mately the photovoltaic production in TWh of the coun-try E . 2 The total photovoltaic production of 28 European Union countries in 2019 of 131.8 TWh . a Show that the countries A and B together account for around 54 % of European production. b Total photovoltaic production in the 28 countries of the European Union was 122.3 TWh in 2018 . What is the percentage increase in total photovoltaic production between 2018 and 2019 ? Round the result to the nearest tenth. 3 We want to study in the country D the evolution of elec-tricity production by type of energy from 2017 ) 2019 . We then use the spreadsheet to produce the following table : a Name the types of energy whose production has in-creased each year from 2017 to 2019 . b What formula could have been entered in cell B9 before stretching it to cell D9 . 8. Unclassified exercises E.10911 Some friends live in Strasbourg and are planning their vacation. This year they’ve decided to explore a major French city for a week. To get there, they’re renting a car. Once there, they will then make all their journeys on foot or by public transport. A car rental agency offers the following three formulas for a one-week rental: Indicative table of distances (in km) between French cities Example : the shortest distance between Nantes and Greno-ble is 741 km . The friends wish to travel to Marseille. They have a budget of 1 000 e for the trip. 1 How far, in km , will they travel for the round trip? 2 Choosing the formula B , show that car hire will cost 701.50 e . 3 Which is the most advantageous formula? 4 Here’s some travel information : Will their budget be sufficient? In this question, any trace of research will be taken into account in the correction. https://chingmath.fr chapExoCorrec/10893 sacados/10893 Pays APays BPays CPays DPays E05101520253035404550 123456789ABCDTyped’énergieNucléaireThermique(gas,∏oulcharbon)HydrauliqueEolienSolaireBioénergiesTotalen2017379,153,953,524,19,29,5529,3Productionélectrique(enTWh)en2018393,239,468,327,810,29,7548,6en2019379,542,66034,111,69,9537,7 chapExoCorrec/10911 sacados/10911 Asie Juin 2024 FormuleAFormuleBFormuleC0;50epourchaque kilomètreparcouruForfait ∏xe300epuis0;25epourchaque kilomètreparcouruForfait ∏xe de900epour unkilométrageillimité GrenobleLilleMarseilleNantesParisStrasbourgBordeaux6757925553385469077712807415855061005584215498909772803379864442 Information1Information2Information3Prix moyen dugazole en 20231;87epar litreVoitureproposéeType decarburant:gazole.Consommation:5;6‘pour100kmCoût totalpour les péages115;80e
Class40OceanFiftyImocaUltim404372054600milles5800milles5800milles7500millesRedmanPrimonialLinkedOutMaxiEdmondde Rothschild21 jours22 heures33 minutes15 jours13 heures27 minutes18 jours1 heures21 minutes16 jours1 heures48 minutesNombresdebateauxdelacatégorieDistanceduparcoursNomdubateauvainqueurdelacatégorieDuréedecourseduvainqueur 12345678910111213141516ABCDEFNationsClasse-mentOrArgentBronzeTotalChine1966051207Grande-Bretagne2413845124États-Unis3373631104Comitéparalym-piqueRusse4363349118Pays-Bas525171759Ukraine624472798Brésil722203072Australie821293080Italie9142969Azerbaïdjan10141419Japon1113152351Allemagne1213121843Iran131211124France1411152854Espagne159151236 E.10915 The Transat Jacques Vabre is a sailing race that connects the city of Le Havre, in mainland France, to the city of Fort-de-France, in Martinique. In the 2021 edition, 75 boats took part in this race, divided into four categories depending on the course to be completed : Class 40, Ocen Fifty, Imoca, Ultim. The table below shows the categories, the number of partici-pants, the distances covered, and the Transat winners : Information : A nautical mile is a unit of measurement used in sailing, equiv-alent to approximately 1.852 km . 1 Show that the boat LinkedOut took 2 days, 11 hours, and 54 minutes longer than the Primonial boat to complete the course. 2 Calculate the average distance traveled by all 75 boats. Round this distance to the nearest whole number. 3 The average speed of the Redman boat was approxi-mately 8.7 milles = h . Show that the average speed of the Maxi Edmond de Rothschild boat was approximately 2.2 times greater than that of the Redman boat. 4 A journalist claims that the distance traveled by an Ocean Fifty class boat is approximately equal to a quar-ter of the circumference of the Earth’s equator. Given that the radius of the equator is 6 370 km , is the journalist correct? Hint: we will use : ı ≈ 3.1416 E.10916 During the Paralympic Games in 2021 , the media ranked countries based on the distribution of medals won. Here are the rankings for the top 15 countries : 1 How many silver medals did Australia win? 2 Calculate the number of bronze medals won by Italy. 3 What formula could have been entered in F2 before being stretched downwards? 4 For each of the following statements, indicate whether it is true or false. Remember that answers must be justified. Statement 1: ˇ 20 % of the medals won by the French team are gold. ı Statement 2: ˇ The median number of silver medals won by these 15 countries is 29 . ı 5 At the Paralympic Games in Rio in 2016 , the bonus for a French gold medal was 50 000 euros. For those in Tokyo in 2021 , this bonus was 65 000 euros. What is the percentage increase in this bonus between 2016 and 2021 ? https://chingmath.fr chapExoCorrec/10915 sacados/10915 Class40OceanFiftyImocaUltim404372054600milles5800milles5800milles7500millesRedmanPrimonialLinkedOutMaxiEdmondde Rothschild21 jours22 heures33 minutes15 jours13 heures27 minutes18 jours1 heures21 minutes16 jours1 heures48 minutesNombresdebateauxdelacatégorieDistanceduparcoursNomdubateauvainqueurdelacatégorieDuréedecourseduvainqueur chapExoCorrec/10916 sacados/10916 Asie Juin 2022 12345678910111213141516ABCDEFNationsClasse-mentOrArgentBronzeTotalChine1966051207Grande-Bretagne2413845124États-Unis3373631104Comitéparalym-piqueRusse4363349118Pays-Bas525171759Ukraine624472798Brésil722203072Australie821293080Italie9142969Azerbaïdjan10141419Japon1113152351Allemagne1213121843Iran131211124France1411152854Espagne159151236
E.10917 In a home, water consumption can be abnormally high when there is a water leak. Consider the following situation : A bathroom is equipped with a cylindrical sink, as shown in the image below. The faucet leaks at a rate of one drop per second. On average, 20 drops of water correspond to one milliliter ( 1 m‘ ) . Characteristics of the basin: Inner diameter : 40 cm Inner height : 15 cm Mass : 25 kg Reminders: Volume of the cylinder: ı × rayon 2 × hauteur 1 dm 3 = 1 ‘ Indication : we will use : ı ≈ 3.1416 1 Due to the leak, show that 86 400 drops fall into the basin in a full day. 2 Calculate, in liters, the volume of water that falls into the basin in a week due to the leak. 3 Show that the basin has a volume of 18.85 liters, rounded to the nearest centiliter. 4 The basin drain is closed and the apartment is unoccu-pied for a week. Will the water overflow from the basin? Justify your answer. 5 At the end of the XIX e century, domestic water consump-tion per capita in France was approximately 17 liters per day. It increased significantly with the widespread distri-bution of tap water in homes, rising to 165 liters per day per capita in 2004 . In 2018 , French consumption fell slightly to 148 liters of water per day per capita. Calculate the percentage decrease in daily water con-sumption per capita between 2004 and 2018 . Round this percentage to the nearest whole number. https://chingmath.fr chapExoCorrec/10917 sacados/10917