# Solution of Chapter 21. Surface Area and Volume of a Sphere (RD Sharma - Mathematics Book)

## Exercise 21.1

1

Find the Find the surface area of a sphere of radius :

(i) 10.5 cm

(ii) 5.6 cm

(iii) 14 cm

2

Find the surface area of a sphere of diameter :

(i) 14 cm

(ii) 21 cm

(iii) 3.5 cm

3

Find the total surface area of a hemisphere and a solid hemisphere each of radius 10 cm. (Use π = 3.14)

4

The surface area of a sphere is 5544 cm2, find its diameter.

5

A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of the plating it on the inside at the rate of Rs. 4 per 100 cm2.

6

The dome of a building is in the form of a hemisphere. Its radius is 63 dm. Find the cost of painting it at the rate of Rs. 2 per sq. m.

7

Assuming the earth to be a sphere of radius 6370 km, how many square kilometres is area of the land, if three-fourth of the earth’s surface is covered by water?

8

A cylinder of same height and radius is placed on the top of a hemisphere. Find the curved surface area of the shape if the length of the shape is 7 cm.

9

A wooden toy is in the form of a cone surmounted on a hemisphere. The diameter of the base of the cone is 16 cm and its height is 15 cm. Find the cost of painting the toy at Rs. 7 per 100 cm2.

10

A storage tank consists of a circular cylinder with a hemisphere adjoined on either end. If the external diameter of the cylinder be 1.4 m and its length be 8 m, find the cost of painting it on the outside at the rate of Rs. 10 per m2.

11

The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.

12

A hemi-spherical dome of a building needs to be painted. IF the circumference of the base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is Rs. 5 per 100 cm2.

13

The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in Fig. 21.11. Eight such spheres are used for this purpose, and are to be painted silver. Each support is a cylinder of a radius 1.5 cm and height 7 cm and is to be painted black. Find the cost of paint required if silver paint costs 25 paise per cm2 and black paint cost 5 paise per cm2. ## Exercise 21.2

1

Find the volume of a sphere whose radius is :

(i) 2 cm

(ii) 3.5 cm

(iii) 10.5 cm

2

Find the volume of a sphere whose diameter is:

(i) 14 cm

(ii) 3.5 dm

(iii) 2.1 m

3

A hemispherical tank has inner radius of 2.8 m. Find its capacity in litres.

4

A hemispherical bowl is made of steel 0.25 cm thick. The inside radius of the bowl is 5 cm. Find the volume of steel used in making the bowl.

5

How many bullet can be made out of a cube of lead, whose edge measures 22 cm, each bullet being 2 cm in diameter?

6

7

A spherical ball of lead 3 cm in diameter is melted and recast into three spherical balls. If the diameters of two balls be cm and 2 cm, find the diameter of the third ball.

8

A sphere of radius 5 cm is immersed in water filled in a cylinder, the level of water rises cm. Find the radius of the cylinder.

9

If the radius of a sphere is doubled, what is the ratio of the volume of the first sphere to that of the second sphere?

10

A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights.

11

A vessel in the form of a hemispherical bowl is full of water. Its contents are emptied in a right circular cylinder. The internal radii of the bowl and the cylinder are 3.5 cm and 7 cm respectively. Find the height to which the water will rise in the cylinder.

12

A cylinder whose height is two thirds of its diameter, has the same volume as a sphere of radius 4 cm. Calculate the radius of the base of the cylinder.

13

A vessel in the form of a hemispherical bowl is full of water. The contents are emptired into a cylinder. The internal radii of the bowl and cylinder are respectively 6 cm and 4 cm. Find the height of water in the cylinder.

14

A cylindrical tub of radius 16 cm contains water to a depth of 30 cm. A spherical iron ball is dropped into the cylinder and thus the level of water is raised by 9 cm. Find the radius of the ball. (Use π =22/7).

15

A cylinder of radius 12 cm contains water to a depth of 20 cm. A spherical iron ball is dropped into the cylinder and thus the level of water is raised by 6.75 cm. Find the radius of the ball (Use π= 22/7)

16

The diameter of a copper sphere is 18 cm. The sphere is melted and is drawn into a long wire of uniform circular cross-section. If the length of the wire is 108 m, find its diameter.

17

A cylindrical jar of radius 6 cm contains oil. Iron spheres each of radius 1.5 cm are immersed in the oil. How many spheres are necessary to raise the level of the oil by two centimetres?

18

A measuring jar of internal diameter 10 cm is partially filled with water. Four equal spherical balls of diameter 2 cm each are dropped in it and they sink down in water completely. What will be the change in the level of water in the jar?

19

The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 0.2 cm. Find the length of the wire.

20

The radius of the internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respectively. If it is melted and recast into a solid cylinder of height cm. Find the diameter of the cylinder.

21

A hemisphere of lead of radius 7 cm is cast into a right circular cone of height 49 cm. Find the radius of the base.

22

A hollow sphere of internal and external radii 2 cm and 4 cm respectively is melted into a cone of base radius 4 cm. Find the height and slant height of the cone.

23

A metallic sphere of radius 10.5 cm is melted and thus recast into small cones, each of radius 3.5 cm and height 3 cm. Find how many cones are obtained.

24

A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights.

25

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1 : 2 : 3.

26

A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical form ball is dropped into the tub and thus the level of water is raised by 6.75 cm. What is the radius of the ball?

27

The largest sphere is carved out of a cube of side 10.5 cm. Find the ratio of their volumes.

28

A sphere, a cylinder and a cone have the same diameter. The height of the cylinder and also the cone are equal to the diameter of the sphere. Find the ratio of their volumes.

29

A cube of side 4 cm contains a sphere touching its side. Find the volume of the gap in between.

30

A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.

31

A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm3) is needed to full this capsule?

32

The diameter of the moon is approximately one fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

## CCE - Formative Assessment

1

Find the surface area of a sphere of radius 14 cm.

2

Find the total surface area of a hemisphere of radius 10 cm.

3

Find the radius of a sphere whose surface area is 154 cm2.

4

The hollow sphere, in which the circus motor cyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding.

5

Find the volume of a sphere whose surface area is 154 cm2.

6

How many spherical bullets can be made out of a solid cube of lead whose edge measures 44 cm, each bullet being 4 cm in diameter?

7

If a sphere of radius 2r has the same volume as that of a cone with circular base of radius r, then find the height of the cone.

8

If a hollow sphere of internal and external diamaters 4 cm and 8 cm respectively melted into a cone of base diameter 8 cm, then find the height of the cone.

9

The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height of the cone.

10

If a sphere is inscribed in a cube, find the ratio of the volume of cube to the volume of the sphere.

1

In a cone the number of faces is

2

The total surface area of a hemisphere of radius r is

3

The ratio of the total surface area of a sphere and a hemisphere of same radius is

4

A sphere and a cube are of the same height. The ratio of their volumes is

5

The largest sphere is cut off from a cube of side 6 cm. The volume of the sphere will be

6

A cylindrical rod whose height is 8 times of its radius is melted and recast into spherical balls of same radius. The number of balls will be

7

If the ratio of volumes of two spheres is 1: 8, then the ratio of their surface areas is

8

If the surface area of a sphere is 144πm2, then its volume (in m3) is

9

If a solid sphere of radius 10 cm is moulded into 8 spherical solid balls of equal radius, then the surface area of each ball (in sq. cm) is

10

The ratio between the volume of a sphere and volume of a circumscribing right circular cylinder is

11

If a sphere is inscribed in a cube, then the ratio of the volume of the sphere to the volume of the cube is

12

If a solid sphere of radius r is melted and cast into the shape of a solid cone of height r, then the radius of the base of the cone is

13

A sphere is placed inside a right circular cylinder so as to touch the top, base and lateral surface of the cylinder. If the radius of the sphere is r, then the volume of the cylinder is