Grade 9 / Sections, cones and spheres 38 exercises (100% corrected)

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ACDBEFGH ABCDEFGHMNOP3cm3cm ABCDEFGHIJKL ACDEFGHPQMNB 1. Section of the parallelepiped by a plane parallel to an edge E.4206 Consider the cube ABCDEFGH shown below : Determine the nature of the quadrilateral DBFH . E.2945 In space, consider the cube ABCDEFGH of edge 5 cm ; a cut is made parallel to the edge [ AB ] to obtain the section MNOP : 1 Give the exact value of segment MP . 2 Draw the full-size section MNOP . E.5443 Consider the parallelepiped ABCDEFGH with dimen-sions : AB = 6 cm ; BC = 3 cm BF = 2 cm A plane parallel to the edge [ FG ] intercepts the paral-lelepiped forming for section the quadrilateral IJKL où : AI = 3 cm ; KG = 2 cm 1 What is the nature of the quadrilateral? 2 a Determine the measure, rounded to the nearest mil-limetre, of the length IL . b Determine the area of the quadrilateral IJKL rounded to the nearest square centimeter. E.6453 Consider a cube ABCDEFGH with edge length 6 cm . Let M , N , P , Q be the respective mid-points of edges [ BF ] , [ CG ] , [ DC ] , and [ AB ] . Here is a representation (that is not to scale) of this configu-ration: 1 Determine the length of segment [ MQ ] to the nearest millimeter. 2 a Determine the volume of the right prism PCNQBM . b Deduce the volume of the solid DPNGHAQMFE . https://chingmath.fr chapExoCorrec/4206 sacados/4206 Polynesie Juin 2010 ACDBEFGH chapExoCorrec/2945 sacados/2945 ABCDEFGHMNOP3cm3cm chapExoCorrec/5443 sacados/5443 ABCDEFGHIJKL chapExoCorrec/6453 sacados/6453 ACDEFGHPQMNB
ACDABCDPRMNB ABCDEFGHIJK ABCDEFGH24cm11;7cm4;4cm ABCDOOH E.4205 The cube shown here is a 6 cm edge cube. (the figure is not of actual dimensions) We consider : the point M middle of the edge [ BB ] ; the point N middle of the edge [ CC ] ; the point P middle of the edge [ DC ] ; point R middle of the edge [ AB ] . 1 What is the nature of the triangle BRM ? Construct this triangle to full size. Calculate the exact value of RM . 2 The cube is cut by the plane passing through R and par-allel to the edge [ BC ] . The section is the quadrilateral RMNP . What is the nature of the section RMNP ? Construct RMNP to full size. Give its exact dimensions. 3 Calculate the area of the triangle RBM . Calculate the volume of the right prism with base the triangle RBM and height [ BC ] . 2. Other sections of the parallelepiped E.7980 Consider the cube ABCDEFGH shown below and the pyramid KIJF with vertex F obtained by sectioning the cube where the points I , J , K belong to the segments [ FE ] , [ FB ] , [ FK ] respectively: Give the nature of each face of the pyramid KIJF . E.7992 Consider the parallelepiped ABCDEFGH whose dimensions are shown in the represen-tation below : A section of this solid is made to obtain the triangle EGB . Determine the dimensions of the triangle BEG . 3. Section of the cylinder E.2946 In space, consider a cylinder of revolution with axis ( OO ) , ra-dius 5 cm , and height 3 cm ; the section of this cylinder by a plane parallel to the axis of revolution gives the quadrilat-eral ABCD ; The distance HO measures 2 cm . 1 a Draw the triangle O BC to scale. https://chingmath.fr chapExoCorrec/4205 sacados/4205 Amerique du Sud Novembre 2009 ACDABCDPRMNB chapExoCorrec/7980 sacados/7980 ABCDEFGHIJK chapExoCorrec/7992 sacados/7992 ABCDEFGH24cm11;7cm4;4cm chapExoCorrec/2946 sacados/2946 ABCDOOH
ABCDEFGHS ABCDEFGIOIH PQ Figure 1Figure 2 b Determine the length BC , rounded to the nearest millimeter. 2 a Give the nature of the quadrilateral ABCD . b Represent the section ABCD to scale. 4. Section of a pyramid E.7977 Consider the pyramid ABCDS with vertex S and base rectangular : We cut this solid by a plane ( P ) parallel to the base. We get the EFGH section. 1 What is the nature of section EFGH ? 2 What is the nature of the solid EFGHS ? E.7978 Consider a pyramid ABCDO with square base ABCD shown opposite. Note I the foot of the height of the pyramid and E the middle of [ OA ] . We have the dimensions : AB =3 cm ; IO =4 cm The plane parallel to the base passing through the point E intercepts the pyramid, forming the quadrilateral EFGH . 1 What is the nature of the quadrilateral EFGH . 2 Justify that the point F is the midpoint of the segment [ OB ] . 3 Draw the base of pyramid EFGHO , vertex O , full size. 5. Section of the sphere E.5674 Consider the sphere and the two cylinders shown below : 1 What is the nature of the cross-section of the sphere with the ( P ) plane? 2 What is the nature of the section of the first cylinder with the plane ( Q ) that is perpendicular to the cylinder’s axis of revolution? 3 What is the nature of the cross-section of the second cylinder with the ( R ) plane that is parallel to the cylin-der’s axis of revolution? 6. Section of a cone of revolution E.7982 Below, we consider the two sections of cones of revolution : https://chingmath.fr chapExoCorrec/7977 sacados/7977 ABCDEFGHS chapExoCorrec/7978 sacados/7978 ABCDEFGIOIH chapExoCorrec/5674 sacados/5674 PQ chapExoCorrec/7982 sacados/7982 Figure 1Figure 2
AOH AIOP ORHM(P 1 In Figure 1 , the section is through a plane parallel to the base of the cone of revolution. What is the nature of the section? 2 In Figure 2 , the section is made by a plane perpendicular to the base of the cone of revolution and passing through the top of the cone. What is the nature of the section? 7. Study of the section of the sphere E.5460 La figure below repre-sents the section of a sphere S by a plane ( P ) . The resulting sec-tion circle is C . 1 What can be said about the points O , H and A shown in the figure below? 1 What do the lengths AH , OA and OH each represent? 2 What is the nature of the triangle OHA ? E.809 Let S be a sphere of radius 12 m and a plane P located at a distance of 7 m from the center of the sphere. 1 Justify that the lines ( AI ) and ( OI ) are perpendicular to each other. 2 Relative to the sphere and circle-section, what do each of the lengths OI , IA and OA represent. 3 Determine the measure of the radius of the circle-section to the nearest decimeter. E.807 Let S be a sphere with center O and radius 7 cm . Let C be a circle associated with the plane ( P ) such that O is at a distance of 5 cm from the plane ( P ) . Let H be the center of C and M a point in space such that M ∈ C . 1 Draw a diagram showing the sphere to scale as well as the section circle. 2 What can be said about triangle OHM ? 3 Draw triangle OHM to scale. 4 Give the exact value of HM , then its value rounded to the nearest millimeter. E.5458 This exercise is a multiple-choice questionnaire (MCQ) . No justification is required. For each question, three answers are offered, only one answer is correct. No points will be taken off for incorrect answers. For each of the 3 questions, indicate on your copy the question number and copy the correct answer. To answer the questions, observe the figure below : O is the center of the sphere, plane P intersects the sphere along a circle of center H , M is a point of this circle, R is the middle of [ OH ] . 1 The point R ap-partient. . . à the sphere of cen-ter O and ra-dius OM à the ball of cen-ter O and ra-dius OM au plane P 2 the distance from point O to plane P est. . . OM OR OH 3 Si OM =11.7 cm et HM =10.8 cm , alors OH = : : : 4.5 cm 1.2 cm 20.25 cm 8. Volume of the sphere https://chingmath.fr chapExoCorrec/5460 sacados/5460 AOH chapExoCorrec/809 sacados/809 AIOP chapExoCorrec/807 sacados/807 chapExoCorrec/5458 sacados/5458 ORHM(P
2cm RTourdetête HhAOK E.9317 Here are the dimensions of four solids : A pyramid with a height of 6 cm , whose base is a rectan-gle with a length of 6 cm and a width of 3 cm . A cylinder with a radius of 2 cm and a height of 3 cm . A cone with a radius of 3 cm and a height of 3 cm . A sphere with a radius of 2 cm . 1 a Represent the first three solids approximately as in the example opposite : b Place the given dimensions on the representations 2 Classify these four solids in ascend-ing order of volume. Some formulas: 4 3 × ı × rayon 3 ı × rayon 2 × hauteur 1 3 × ı × rayon 2 × hauteur 1 3 × aire de la base × hauteur Hint: we will use : ı ≈ 3 ; 1416 E.2652 In a cubic box whose edge measures 7 cm , we place a 7 cm ball with diameter (see diagram opposite) . The volume of the ball corre-sponds to a certain percent-age of the volume of the box. We call this percentage ˇ fill-ing ratio of the boîte ı. Calculate this filling rate of the box. Round this percentage to the nearest integer. Hint: we’ll use : ı ≈ 3.1416 E.9320 Guillaume would like to know how many hairs he has on his head. To do this, he represents his head as a sphere with ra-dius R . He measures the circumference of his head as shown in the diagram below and obtains 56 cm . Reminders: Perimeter of a circle with radius R : P = 2 ıR Area of a sphere with radius R : A = 4 ıR 2 . Hint: we will use : ı ≈ 3 ; 1416 1 Show that the radius of a circle with perimeter 56 cm is approximately equal to 9 cm . 2 Guillaume estimates that his hair covers half the surface area of his head. On 1 cm 2 of his skull, he counted 250 hairs. Estimate the number of hairs Guillaume has. Hint: for this question, all traces of research will be taken into account when grading. E.4336 1 Draw a right paving block in cavalier perspective. 2 An aquarium has the shape of a right cube of length 40 cm , width 20 cm and height 30 cm . a Calculate the volume, in cm 3 , of this right block. b Remember that a liter corresponds to 1 000 m 3 . How many liters of water can this aquarium hold? Ano justification is required . 3 From the following formulae, copy the one that gives the volume, in cm 3 , of a ball of 30 cm de diameter : 4 3 × ı × 30 3 ; 4 ı × 15 2 ; 4 3 × ı × 15 3 4 A second aquarium contains a volume of water equal to three-quarters the volume of a ball of diameter 30 cm . Its contents are poured into the first aquarium. How high does the water rise? Give a value rounded to the nearest mil-limetre. Indication : we will use : ı ≈ 3.1416 9. Sphere section and volume E.2491 A spherical cap is a solid ob-tained by cutting a sphere with a plane. A liquid detergent dispenser, shown below, has the shape of a spherical cap with center O and radius R = OA =4.5 cm . https://chingmath.fr chapExoCorrec/9317 sacados/9317 2cm chapExoCorrec/2652 sacados/2652 Bordeaux - Juin 2002 chapExoCorrec/9320 sacados/9320 RTourdetête chapExoCorrec/4336 sacados/4336 chapExoCorrec/2491 sacados/2491 HhAOK
OHRTPartie enfouieSolPartie visible(calotte sphérique) 6cm2cm6cm The opening of this container is delimited by the circle with center H and radius HA =2.7 cm . The total height of this measuring device is HK . 1 Draw the triangle AHO to scale. 2 Calculate OH , giving your reasoning, then deduce that the total height HK of the measuring cup is exactly 8.1 cm . 3 The volume V of a spherical cap with radius R and height h is given by the formula : V = 1 3 · ı · h 2 · 3 · R − h Calculate based on ı the exact volume of the dispenser in cm 3 . Deduce the total capacity of the dispenser rounded to the nearest milliliter. Note: we will use : ı ≈ 3.1416 E.5442 To attract more visitors to his city, a mayor decides to build the Pacific Aquarium. The architects plan to install a huge aquarium at the entrance, with spherical glass. The figure below shows the situation. This figure is not to scale. Reminder : the formula for the volume of a sphere with radius R : V boule = 4 × ı × R 3 3 Note: we will use : ı ≈ 3.1416 1 Calculate the volume of a sphere with radius 5 m , rounded to the nearest cubic meter. 2 In reality, the aquarium is built into the ground. The up-per part (visible to visitors) is a ˇ spherical dome ı. The lower part (buried) houses the machinery. a What is the geometric nature of the section between the horizontal plane of the ground and the aquarium (the grayed-out part in the figure) ? b Point O designates the center of the sphere. The fol-lowing actual dimensions are given : OH = 3 m ; RO = 5 m ; HR = 4 m où H and R are the points located on the floor as shown in the figure. Is triangle OHR a right triangle?? Justify your an-swer. 3 a T is a point on the sphere such that points T , O , and H are aligned as shown in the figure. Calculate the height HT of the visible part of the aquarium. b Calculate the volume of this spherical cap, rounded to the nearest liter. Hint: the volume of a spherical cap with radius 5 m is given by the formula : V calotte = ı × h 2 3 × (15 − h ) où h denotes its height (corresponding to the length HT in the figure.) c For this question, we will take the volume of the aquar-ium to be 469 000 liters. Pumps deliver seawater at a constant rate to fill the empty aquarium. In 2 hours of operation, the pumps together inject 14 000 liters of seawater. After how many hours of operation will the pumps have filled the aquarium?? 10. Volumes of solids E.5925 Reminders: Volume of a cylinder: V = ı × rayon 2 × hauteur Volume of a sphere : V = 4 3 × ı × rayon 3 Note: we will use : ı ≈ 3 ; 1416 Flora makes bracelets out of modeling clay. They are all made up of 8 round beads and 4 long beads. This modeling clay is sold in blocks, all of which are shaped like rectangular prisms with the dimensions specified opposite. The clay can be kneaded as de-sired and then hardens when baked. Information about the beads: https://chingmath.fr chapExoCorrec/5442 sacados/5442 OHRTPartie enfouieSolPartie visible(calotte sphérique) chapExoCorrec/5925 sacados/5925 6cm2cm6cm
23cm23cm ABCDEFGHIKLJOS A round bead Ball with a diameter of 8 mm A long bead Cylinder with a height of 16 mm and a diameter of 8 mm Flora buys two blocks of modeling clay: a block of blue mod-eling clay to make round beads and a block of white modeling clay to make long beads. How many bracelets can she expect to make? E.9319 The large crystal globe is a trophy awarded to the winner of the Ski World Cup. This trophy weighs 9 kg and measures 46 cm in height. This globe is considered to be com-posed of a crystal cylinder with a di-ameter of 6 cm and a height of 23 cm , topped with a crystal ball with a di-ameter of 23 cm . See diagram oppo-site. Note: we will use : ı ≈ 3 ; 1416 1 Show that the volume of the ball of this trophy is 6 371 cm 3 , rounded to the nearest cubic centimeter. 2 Marie claims that the volume of the crystal ball repre-sents approximately 90 % of the total volume of the tro-phy. Is she right? E.4203 Consider the following three solids : the ball of center O and radius SO tel that SO =3 cm ; the pyramid SEFGH of height 3 cm whose base is the square EFGH de side 6 cm ; le cube ABCDEFGH d edge 6 cm . These three solids are placed in a container. This container is represented by the right block ABCDIJKL of height 15 cm whose base is the square ABCD de side 6 cm . 1 Calculate the volume of the cube ABCDEFGH en cm 3 . 2 Calculate the volume of the pyramid SEFGH en cm 3 . 3 Determine the volume occupied by the three solids inside the ABCDIJKL cube, rounded to the cm 3 . 4 In this question, write down all calculations to justify your answer. Any trace of research, even if incomplete, will be taken into account in the assessment. Can we pour 20 c‘ water into this container without it overflowing? Diagram : The figure is not to scale. Some indications: The volume of a pyramid is calculated using the for-mula : V = 1 3 × h × B where h is the height of the pyramid and B is the area of its base. The volume of a sphere is calculated using the formula : V = 4 3 × ı × r 3 where r is the radius of the sphere. 1 dm 3 = 1 ‘ we will use : ı ≈ 3.1416 11. Geographical coordinates on the sphere E.7001 Below are the meridians and paral-lels of the globe : https://chingmath.fr chapExoCorrec/9319 sacados/9319 23cm23cm chapExoCorrec/4203 sacados/4203 ABCDEFGHIKLJOS chapExoCorrec/7001 sacados/7001
N-80-60-40-200204060-120-100-40-20020406080ABC NS0o40oS65oN0o30oW150oE50oEABCD 15o30o45o60o75o90o105o120o135o150o165o180o165o0oNordSud15o30o45o60o75o90o105o120o135o150o165o0oéquateurEstOuest15o30o45o60o75o15o30o45o60o75oPyeongchang NSABCD ABSNOGEquateur Determine the geodesic coordinates of points A , B , and C . E.2943 On the sphere below representing the earth, consider the points A , B , C , D shown below : Read the geographic coordinates of these four points. E.9318 French biathlete Martin Four-cade won the sixth big crystal globe of his career in 2017 Pyeongchang, South Korea. Give approximately the latitude and longitude of this location marked on the map below : E.2944 We consider on the earth, four points A , B , C , D where we know the ge-ographical coordinates of the points A and B : A : 30 o S 10 o W B : 55 o N 130 o E Determine the geographic coordinates of the points C and D . 12. Geographical coordinates and distance calculation E.2653 The earth is assimilated to a sphere of radius 6 370 km . 1 Consider the plane perpendicular to the pole line ( NS ) and equidistant from these two poles. The intersection of this plane with the earth is called the equator. Calculate the length of the equator rounded to the near-est kilometer. 2 Note O the center of the earth and G a point on the equator. Consider two points A and B located in Africa on the equator. These points are arranged as shown in the dia-gram above. We know that : GOA =42 o ; GOB =9 o . Calculate the length of the arc AB , the portion of the equator located in Africa rounded to the nearest kilome-ter. https://chingmath.fr N-80-60-40-200204060-120-100-40-20020406080ABC chapExoCorrec/2943 sacados/2943 NS0o40oS65oN0o30oW150oE50oEABCD chapExoCorrec/9318 sacados/9318 15o30o45o60o75o90o105o120o135o150o165o180o165o0oNordSud15o30o45o60o75o90o105o120o135o150o165o0oéquateurEstOuest15o30o45o60o75o15o30o45o60o75oPyeongchang chapExoCorrec/2944 sacados/2944 NSABCD chapExoCorrec/2653 sacados/2653 Grenoble - Juin 2002 ABSNOGEquateur
PVI49oO LSOM KEGCOE E.2541 The table below shows three cities: their latitude and longitude. Latitude Longitude Douala 4 o 9 o Cayenne 4 o − 52 o Milan 45 o 9 o Remember that the radius of the Earth is approximately 6 370 km 1 a Determine the distance as the crow flies between Douala and Milan (city in Italy) rounded to the near-est kilometer. b Determine the distance as the crow flies between Douala and Cayenne (city in French Guiana) rounded to the nearest kilometer. 2 The distance as the crow flies from Cayenne to Milan is 7 470 km . Can this distance be deduced from the data obtained in the previous questions? Note: Spherical geometry is said to be a geometry with positive curvature. Hint: we will use : ı ≈ 3.1416 E.2954 Vancouver, Canada, has the following geographical coordinates : 123 o E 49 o N Paris has the following ge-ographical coordinates : 2 o E 49 o N Point I is the only point on the equator such that triangle OIP is a right tri-angle. Note: The radius of the Earth is approximately 6 371 km . We will use : ı ≈ 3.1416 1 Justify that the cities of Vancouver and Paris are on the same latitude. 2 a Determine the radius of the circle defining the lati-tude 49 o N , rounded to the nearest kilometer. b Determine the length of the latitude 49 o N , rounded to the nearest kilometer. c Deduce the length, on the earth’s surface, separating these two cities, rounded to the nearest kilometer. 3 Give the coordinates of the geographical point diametri-cally opposite the city of Vancouver. E.2456 The drawing below represents the Earth, which is assimilated to a sphere of 6 370 km radius. The circle with center O passing through M represents the equator. The point L represents the city of London. L lies on the sphere and on the circle with center S (see figure) . We’ll admit that the angle LSO is a right angle. We give OS =4 880 km . 1 Calculate SL to the nearest kilometer. 2 Calculate the measure of the angle SOL rounded to the nearest degree. 3 Deduce London’s northern latitude from the equator to the nearest degree, i.e. the angle LOM . E.2540 The Earth is considered to be spher-ical; a representation is given opposite, showing : The city G of Greenwich and its meridian The city K of Kiev and its meridian passing through this city. The cities G and K are on the same parallel. Reminder : the radius of the Earth is approximately 6 370 km . Note: we will use : ı ≈ 3.1416 1 Justify that angles GOE and KOE are equal in length. 2 Determine the radius of the circle section formed by the Greenwich parallel, rounded to the nearest kilometer, given that the latitude of Greenwich is 51 o . 3 Determine the distance between Greenwich and Kiev , rounded to the nearest kilometer, given that the longi-tude of Kiev is 31 o W 13. Share https://chingmath.fr chapExoCorrec/2541 sacados/2541 chapExoCorrec/2954 sacados/2954 PVI49oO chapExoCorrec/2456 sacados/2456 LSOM chapExoCorrec/2540 sacados/2540 KEGCOE
AOH OAB IJKL70oOO ABCDLOABCDABCDLOABCDEFGH E.5459 Reminder : volume of a sphere : V = 4 × ı × R 3 3 Note: we will use : ı ≈ 3.1416 1 Calculate the volume of a sphere with radius R =7 cm , rounded to cm 3 . 2 We cut the sphere with center O and radius OA =7 cm with a plane, shown below. What is the nature of this section? 3 Calculate the exact value of the radius HA of this section, given that OH =4 cm . 14. Unclassified exercises E.985 The small ball has a radius of OA =3 cm and a volume of 36 ı while the large one has a volume of 288 ı . 1 Determine the enlargement coefficient from the small ball to the large ball. 2 Deduce the measure of the radius of the large ball [ OB ] . E.5444 Consider the cylinder of revolu-tion shown opposite, where O and O are the centers of two faces. A plane parallel to the axis of ( OO ) intersects the cylinder: the section forms the quadrilateral IJKL . We have the measurements : OO = 6 cm ; OI = 4 cm KO L = 70 o 1 What is the nature of the quadrilateral IJKL ? 2 a What is the nature of triangle O KL ? b By noting H the foot of the height from O in trian-gle O KL , determine the length of the segment [ KL ] rounded to the nearest millimeter. 3 Determine the area of the quadrilateral IJKL rounded to the nearest square centimeter. E.988 The unit of length is the centimeter, the unit of area is the square centimeter, the unit of volume is the cubic centimeter. Consider the right paving block ABCDA B C D . Note L the point of intersection of segments [ AC ] and [ BD ] . This pavement was excavated by removing the pyramid OABCD of height [ OL ] . We have : DD =5 ; DC =6 ; DA =7 Part A In this part, we have : OL =4 . 1 Construct full-scale, face ABCD and place point L . 2 a Calculate BD (a value rounded to the tenth will be given.) b Deduct DL (we’ll give a value rounded to the tenth) 3 a Calculate the volume of the right block ABCDA B C D . b Calculate the volume of the pyramid OABCD . c Deduce the volume of the hollowed-out paving stone. Part B In this part, we pose OL = x where x is a number between 0 and 5. The resulting hollowed-out paving stone is the wooden base of a trophy. On this base, we place a glass pyramid OEFGH which is an enlargement of the pyramid OABCD , of ratio 2. https://chingmath.fr chapExoCorrec/5459 sacados/5459 AOH chapExoCorrec/985 sacados/985 OAB chapExoCorrec/5444 sacados/5444 IJKL70oOO chapExoCorrec/988 sacados/988 ABCDLOABCDABCDLOABCDEFGH
1 a Calculate the volume of the pyramid OABCD as a function of x . b Show that the volume of the wooden base is : 210 − 14 x . 2 Show that the volume of the glass pyramid OEFGH is 112 x 3 Calculate the value of x for which the volume of glass is equal to 2 times the volume of wood. Part C Consider the functions f and g defined by: f : x ↦→ 210 − 14 x et g : x ↦→ 112 x When x is between 0 and 5, the function f represents varia-tions in wood volume and the function g represents variations in glass volume. 1 Graph the functions f and g for x between 0 and 5. For the benchmark, we’ll take : the origin at the bottom left of the sheet ; on the x-axis 2cm for 1 unit ; on the y-axis 1cm for 25 units. the x-axis and y-axis are perpendicular 2 a We want the volume of wood and the volume of glass to be equal. Using the graph, give an approximate value of x for this to be the case. (show the plot used to answer) . b Find this result by calculation. https://chingmath.fr