The volume of a sphere of radius r is given by one of the formulae below.
A) 4 π r2
B) 4/3 π r3
C) 1/3 π r2 h
D) π r l
(Total for Question 1 is 1 mark)
2
A sphere has radius 6 cm. Calculate the volume of the sphere. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 2 is 2 marks)
3
A cone has base radius 4 cm and perpendicular height 9 cm. Calculate the volume of the cone. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
A sphere has diameter 10 cm. Calculate the surface area of the sphere. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 4 is 3 marks)
5
A cone has base radius 3 cm and slant height 8 cm. Calculate the curved surface area of the cone. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 5 is 2 marks)
6
A cone has base radius 5 cm and slant height 13 cm. Show that the curved surface area of the cone is 65 π cm2.
Diagram NOT accurately drawn
(Total for Question 6 is 2 marks)
7
A cone has base radius 6 cm and slant height 10 cm. Calculate the total surface area of the cone (the curved surface area plus the area of the circular base). Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 7 is 3 marks)
8
A cone has base radius 9 cm and slant height 15 cm.
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(a)Calculate the perpendicular height of the cone.(2)
(b)Hence calculate the volume of the cone. Give your answer correct to 3 significant figures.(2)
(Total for Question 8 is 4 marks)
9
A solid hemisphere has radius 9 cm. Calculate the volume of the hemisphere. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 9 is 2 marks)
10
A solid hemisphere has radius 7 cm. Calculate the total surface area of the hemisphere (the curved surface area plus the flat circular face). Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 10 is 3 marks)
11
A toy is made from a cone of radius 6 cm and height 8 cm fixed on top of a hemisphere of radius 6 cm, so that the flat faces are joined together. Calculate the total volume of the toy. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 11 is 4 marks)
12
A sphere has volume 972 π cm3. Find the radius of the sphere.
(Total for Question 12 is 3 marks)
13
A cone has base radius 5 cm and volume 100 π cm3. Find the perpendicular height of the cone.
(Total for Question 13 is 3 marks)
14
A sphere of radius 8 cm fits exactly inside a cube of side length 16 cm, touching every face of the cube. Work out the volume of empty space inside the cube that is not occupied by the sphere. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 14 is 4 marks)
15
A solid sphere of radius 4 cm is made from metal with density 8.9 g/cm3. Calculate the mass of the sphere in kilograms. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 15 is 3 marks)
16
A cone has base diameter 1.4 m and perpendicular height 90 cm. Calculate the volume of the cone in cubic metres. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 16 is 3 marks)
17
A spherical scoop of ice cream has radius 3 cm. It is placed on top of an empty cone-shaped wafer with radius 3 cm and height 10 cm. If all of the ice cream melts and drips into the wafer, determine whether the wafer will overflow. You must show your working.
Diagram NOT accurately drawn
(Total for Question 17 is 3 marks)
18
A solid dome is made from a cone of radius 8 cm and perpendicular height 15 cm fixed on top of a hemisphere of radius 8 cm, with the flat faces joined together.
Diagram NOT accurately drawn
(a)Calculate the slant height of the cone.(2)
(b)Calculate the total area of the dome's surface that needs to be painted: the curved surface of the cone plus the curved surface of the hemisphere only (do not include the flat circle where the two solids join). Give your answer as a multiple of π.(3)
(c)Paint costs 2p per cm2. Work out the total cost of painting 50 of these dome solids. Give your answer in pounds, to the nearest penny.(3)
(Total for Question 18 is 8 marks)
19
Two similar spherical balloons have radii in the ratio 2:5. The volume of the smaller balloon is 32 cm3. Work out the volume of the larger balloon.
(Total for Question 19 is 3 marks)
20
A solid wooden hemisphere has radius 10 cm. A cone-shaped hole of radius 6 cm and depth 8 cm is drilled out of the flat face, with the apex of the cone pointing into the hemisphere. Work out the volume of wood remaining. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 20 is 4 marks)
21
A large cone has base radius 10 cm and perpendicular height 15 cm. A smaller, similar cone of radius 4 cm and height 6 cm is removed from the vertex end, leaving a frustum. Work out the volume of the frustum. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 21 is 5 marks)
Mark scheme · 5.16 Spheres and Cones
Question 1
B1 B cao
Answer: B) 4/3 π r3
Question 2
M1 4/3 x π x 63 oe (288 π seen)
A1 awrt 905
Answer: 905 cm3 (3 sf)
Question 3
M1 1/3 x π x 42 x 9 oe (48 π seen)
A1 awrt 151
Answer: 151 cm3 (3 sf)
Question 4
B1 radius = 5 cm identified
M1 4 x π x 52 oe (100 π seen)
A1 awrt 314
Answer: 314 cm2 (3 sf)
Question 5
M1 π x 3 x 8 oe (24 π seen)
A1 awrt 75.4
Answer: 75.4 cm2 (3 sf)
Question 6
M1 π x 5 x 13 correctly substituted into curved surface area = π r l
A1 cso: 65 π cm2 obtained exactly, with the answer given (printed) reached correctly
Answer: 65 π cm2 (exact, as given)
Question 7
M1 π x 6 x 10 oe (curved surface area = 60 π)
M1 π x 62 oe (base area = 36 π)
A1 awrt 302
Answer: 302 cm2 (3 sf)
Question 8
(a) M1√152 - 92 oe (225 - 81 = 144 seen)
(a) A1 12 cm cao
(a) Answer: 12 cm
(b) M1 1/3 x π x 92 x their height (ft from part a)
(b) A1 awrt 1020
(b) Answer: 1020 cm3 (3 sf)
Question 9
M1 2/3 x π x 93 oe (486 π seen)
A1 awrt 1530
Answer: 1530 cm3 (3 sf)
Question 10
M1 2 x π x 72 oe (curved surface area = 98 π)
M1 π x 72 oe (flat face area = 49 π)
A1 awrt 462
Answer: 462 cm2 (3 sf)
Question 11
M1 1/3 x π x 62 x 8 oe (cone volume = 96 π)
M1 2/3 x π x 63 oe (hemisphere volume = 144 π)
M1 adds their two volumes (dep)
A1 awrt 754
Answer: 754 cm3 (3 sf)
Question 12
M1 sets 972 π = 4/3 π r3 and begins to rearrange
M1 r3 = 729 oe (dep)
A1 r = 9 cm cao
Answer: r = 9 cm
Question 13
M1 sets 100 π = 1/3 x π x 52 x h and begins to rearrange
M1 h = 100 x 3 / 25 oe (dep)
A1 h = 12 cm cao
Answer: h = 12 cm
Question 14
M1 163 oe (cube volume = 4096)
M1 4/3 x π x 83 oe (sphere volume seen, awrt 2140 to 2150)
M1 subtracts sphere volume from cube volume (dep)
A1 awrt 1950
Answer: 1950 cm3 (3 sf)
Question 15
M1 4/3 x π x 43 oe (volume = 268.08... cm3)
M1 their volume x 8.9 (mass in grams, dep)
A1 awrt 2.39 (kg), correctly converted from grams
Answer: 2.39 kg (3 sf)
Question 16
B1 consistent units used, e.g. radius = 0.7 m and height = 0.9 m
M1 1/3 x π x 0.72 x 0.9 oe (0.147 π seen)
A1 awrt 0.462
Answer: 0.462 m3 (3 sf)
Question 17
M1 4/3 x π x 33 oe (sphere volume = 36 π, awrt 113)
M1 1/3 x π x 32 x 10 oe (cone volume = 30 π, awrt 94.2)
A1 cao: correct conclusion that the wafer will overflow, since 113 > 94.2 (or 36 π > 30 π), with correct comparison shown
Answer: Yes, it overflows (113 cm3 melted ice cream compared with 94.2 cm3 capacity)
Question 18
(a) M1√82 + 152 oe (64 + 225 = 289 seen)
(a) A1 17 cm cao
(a) Answer: 17 cm
(b) M1 π x 8 x their 17 oe (cone curved surface area = 136 π, ft)
(b) M1 2 x π x 82 oe (hemisphere curved surface area = 128 π)
(b) A1 264 π cm2 oe cao
(b) Answer: 264 π cm2
(c) M1 264 π x 50 oe (total area for 50 domes, ft from part b, awrt 41500)
(c) M1 their total area x 0.02 (converts 2p per cm2 correctly to pounds, dep)