## Chapter 19 – Volume and Surface Areas of Solids Exercise Ex. 19A

Two

cubes each of volume 27 cm’ are joined end to end to form a solid. Find the

surface area of the resulting cuboid.

If

the total surface area of a solid hemisphere is 462 cm^{2}, find its

volume.

A

5-m-wide cloth is used to make a conical tent of base diameter 14 m and

height 24 m. Find the cost of cloth used at the rate of Rs. 25 per metre.

If

the volumes of two cones are in the ratio of 1: 4 and their diameters are in

the ratio of 4 : 5, find the ratio of their heights.

The

slant height of a conical mountain is 2.5 km and the area of its base is 1.54

km^{2}. Find the height of the mountain.

The

sum of the radius of the base and the height of a solid cylinder is 37

metres. If the total surface area of the cylinder be 1628 sq metres, find its

volume.

The

surface area of a sphere is 2464 cm^{2}. If its radius be doubled,

what will be the surface area of the new sphere?

A

military tent of height 8.25 m is in the form of a right circular cylinder of

base diameter 30 m and height 5.5 in surmounted by a right circular cone of

same base radius. Find the length of canvas used in making the tent, if the

breadth of the canvas is 1.5 m.

s

A tent is in the shape of a right circular cylinder up to a height of 3 m and conical above it. The total height of the tent is 13.5 m and the radius of its base is 14 m. Find the cost of cloth required to make the tent at the rate of Rs.80 per square meter. Take

Radius of the cylinder = 14 m

And its height = 3 m

Radius of cone = 14 m

And its height = 10.5 m

Let l be the slant height

Curved surface area of tent

= (curved area of cylinder + curved surface area of cone)

Hence, the curved surface area of the tent = 1034

Cost of canvas = Rs.(1034 × 80) = Rs. 82720

A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and the slant height of the conical portion is 53 m, find the area of canvas needed to make the tent. Take .

For the cylindrical portion, we have radius = 52.5 m and height = 3 m

For the conical portion, we have radius = 52.5 m

And slant height = 53 m

Area of canvas = 2rh + rl = r(2h + l)

A rocket is in the form of a circular cylinder closed at the lower end and a cone of the same radius is attached to the top. The radius of the cylinder is 2.5m, its height of 21 m and the slant height of the cone is 8 m. Calculate the total surface area of the rocket.

Radius o f cylinder = 2.5 m

Height of cylinder = 21 m

Slant height of cone = 8 m

Radius of cone = 2.5 m

Total surface area of the rocket = (curved surface area of cone

+ curved surface area of cylinder + area of base)

A

solid is in the shape of a cone surmounted on a hemisphere, the radius of

each of them being 3.5 cm and the total height of the solid is 9.5 cm. Find

the volume of the solid.

A toy is in the form of a cone mounted on a hemisphere of common base radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.

Height of cone = h = 24 cm

Its radius = 7 cm

Total surface area of toy

A

toy is in the shape of a cone mounted on a hemisphere of same base radius. If

the volume of the toy is 231 cm^{3} and its diameter is 7 cm, find

the height of the toy.

A

cylindrical container of radius 6 cm and height 15 cm is filled with ice-cream.

The whole ice-cream has to be distributed to 10 children in equal cones with

hemispherical tops. If the height of the conical portion is 4 times the

radius of its base, find the radius of the ice-cream cone.

A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder. The diameter of the hemisphere is 21 cm and the total height of the vessel is 14.5 cm. Find its capacity.

Radius of hemisphere = 10.5 cm

Height of cylinder = (14.5 10.5) cm = 4 cm

Radius of cylinder = 10.5 cm

Capacity = Volume of cylinder + Volume of hemisphere

A

toy is in the form of a cylinder with hemispherical ends. If the whole length

of the toy is 90 cm and its diameter is 42 cm, find the cost of painting the

toy at the rate of 70 paise per sq cm.

A

medicine capsule is in the shape of a cylinder with two hemispheres stuck to

each of its ends. The length of the entire capsule is 14 mm and the diameter

of the capsule is 5 mm. Find its surface area.

A wooden article was made by scooping out a hemisphere from each end of a cylinder, as shown in the figure. If the height of the cylinder is 20 cm and its base is of diameter 7 cm, find the total surface area of the article when it is ready.

Height of cylinder = 20 cm

And diameter = 7 cm and then radius = 3.5 cm

Total surface area of article

= (lateral surface of cylinder with r = 3.5 cm and h = 20 cm)

A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 21. cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub full of water in such a way that the whole solid is submerged in water. If the radius of the cylinder is 5 cm and its height is 9.8 cm, find the volume of the water left in the tub.

Radius of cylinder

And height of cylinder

Radius of cone r = 2.1 cm

And height of cone

Volume of water left in tub

= (volume of cylindrical tub – volume of solid)

From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8cm and of base radius 6 cm, is hollowed out. Find the volume of the remaining solid. Also, find the total surface area of the remaining solid. Take = 3.14

(i)Radius of cylinder = 6 cm

Height of cylinder = 8 cm

Volume of cylinder

Volume of cone removed

(ii)Surface area of cylinder = 2 = 2× 6 × 8

From

a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of

the same height and same diameter is hollowed out. Find the total surface

area of the remaining solid.

From

a solid cylinder of height 14 cm and base diameter 7 cm, two equal conical

holes each of radius 2.1 cm and height 4 cm are cut off. Find the volume of

the remaining solid.

A spherical glass vessel has a cylindrical neck 7 cm long and 4 cm in diameter. The diameter of the spherical part is 21 cm. Find the quantity of water it can hold. Use = .

Diameter of spherical part of vessel = 21 cm

The adjoining figure represents a solid consisting of a cylinder surmounted by a cone at one end and a hemisphere at the other. Find the volume of the solid.

Height of cylinder = 6.5 cm

Height of cone =

Radius of cylinder = radius of cone

= radius of hemisphere

=

Volume of solid = Volume of cylinder + Volume of cone

+ Volume of hemisphere

From

a cubical piece of wood of side 21 cm, a hemisphere is carved out in such a

way that the diameter of the hemisphere is equal to the side of the cubical

piece. Find the surface area and volume of the remaining piece.

A cubical block of side 10

cm is surmounted by a hemisphere. What is the largest diameter that the

hemisphere can have? Find the cost of painting the total surface area of the

solid so formed, at the rate of Rs.5 per 100 sq cm. [Use π

= 3.14.]

A

toy is in the shape of a right circular cylinder with a hemisphere on one end

and a cone on the other. The radius and height of the cylindrical part are 5

cm and 13 cm respectively. The radii of the hemispherical and the conical

parts are the same as that of the cylindrical part. Find the surface area of

the toy, if the total height of the toy is 30 cm.

The

inner diameter of a glass is 7 cm and it has a raised portion in the bottom

in the shape of a hemisphere, as shown in the figure. If the height of the

glass is 16 cm, find the apparent capacity and the actual capacity of the glass.

A

wooden toy is in the shape of a cone mounted on a cylinder, as shown in the

figure. The total height of the toy is 26 cm, while the height of the conical

part is 6 cm. The diameter of the base of the conical part is 5 cm and that

of the cylindrical part is 4 cm. The conical part and the cylindrical part are

respectively painted red and white. Find the area to be painted by each of

these colours.

## Chapter 19 – Volume and Surface Areas of Solids Exercise Ex. 19B

The

dimensions of a metallic cuboid are 100 cm x 80 cm x 64 cm. It is melted and

recast into a cube. Find the surface area of the cube.

A

cone of height 20 cm and radius of base 5 cm is made up of modelling clay. A

child reshapes it in the form of a sphere. Find the diameter of the sphere.

Metallic

spheres of radii 6 cm, 8 cm and 10 cm respectively are melted to form a

single solid sphere. Find the radius of the resulting sphere.

A solid metal cone with radius of base 12 cm and height 24 cm is melted to form solid spherical balls of diameter 6 cm each. Find the number of balls thus formed.

Radius of the cone = 12 cm and its height = 24 cm

Volume of cone =

The radii of internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respectively. It is melted and recast into a solid cylinder of diameter 14 cm. Find the height of the cylinder.

The internal and external diameters of a hollow hemispherical shell are 6 cm and 10 cm respectively. It is melted and recast into a solid cone of base diameter 14 cm. Find the height of the cone so formed.

Internal radius = 3 cm and external radius = 5 cm

Hence, height of the cone = 4 cm

A

copper rod of diameter 2 cm and length 10 cm is drawn into a wire of uniform

thickness and length 10 m. Find the Thickness of the wire.

A hemispherical bowl of internal diameter 30cm contains some liquid. This liquid is to be filled into cylindrical shaped bottles each of diameter 5 cm and height 6 cm. Find the number of bottles necessary to empty the bowl.

Inner radius of the bowl = 15 cm

Volume of liquid in it =

Radius of each cylindrical bottle = 2.5 cm and its height = 6 cm

Volume of each cylindrical bottle

Required number of bottles =

Hence, bottles required = 60

A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, each of diameter 3.5 cm and height 3 cm. Find the number of cones so formed.

Radius of the sphere=

Let the number of cones formed be n, then

Hence, number of cones formed = 504

A spherical cannon ball 28 cm in diameter is melted and recast into right circular conical mould, base of which is 35 cm in diameter. Find the height of the cone.

Radius of the cannon ball = 14 cm

Volume of cannon ball =

Radius of the cone =

Let the height of cone be h cm

Volume of cone =

Hence, height of the cone = 35.84 cm

A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 1.5cm and 2 cm. Find the radius of third ball.

Let the radius of the third ball be r cm, then,

Volume of third ball = Volume of spherical ball volume of 2 small balls

A spherical shell of lead whose external and internal diameters are respectively 24 cm and 18 cm, is melted ad recast into a right circular cylinder 37 cm high. Find the diameter of the base of the cylinder.

External radius of shell = 12 cm and internal radius = 9 cm

Volume of lead in the shell =

Let the radius of the cylinder be r cm

Its height = 37 cm

Volume of cylinder =

Hence diameter of the base of the cylinder = 12 cm

A hemisphere of lead of radius 9 cm is cast intoa right circular cone of height 72 cm. Find the radius of the base of the cone.

Volume of hemisphere of radius 9 cm

Volume of circular cone (height = 72 cm)

Volume of cone = Volume of hemisphere

Hence radius of the base of the cone = 4.5 cm

A spherical ball of diameter 21 cm is melted and recast into cubes, each of side 1 cm. Find the number of cubes so formed.

Diameter of sphere = 21 cm

Hence, radius of sphere =

Volume of sphere = =

Volume of cube = a^{3} = (1 1 1)

Let number of cubes formed be n

Volume of sphere = n Volume of cube

Hence, number of cubes is 4851.

How many lead balls, each of radius 1 cm, can be made from a sphere of radius 8 cm?

Volume of sphere (when r = 1 cm) = =

Volume of sphere (when r = 8 cm) = =

Let the number of balls = n

A solid sphere of radius 3cm is melted and then cast into small spherical balls, each of diameter 0.6 cm. Find the number of small balls so obtained.

Radius of sphere = 3 cm

Volume of sphere =

Radius of small sphere =

Volume of small sphere =

Let number of small balls be n

Hence, the number of small balls = 1000.

The diameter of a sphere is 42 cm. It is melted and drawn into a cylindrical wire of diameter 2.8 cm. Find the length of the wire.

Diameter of sphere = 42 cm

Radius of sphere =

Volume of sphere =

Diameter of cylindrical wire = 2.8 cm

Radius of cylindrical wire =

Volume of cylindrical wire =

Volume of cylindrical wire = volume of sphere

Hence length of the wire 63 m.

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

Diameter of sphere = 18 cm

Radius of copper sphere =

Length of wire = 108 m = 10800 cm

Let the radius of wire be r cm

But the volume of wire = Volume of sphere

Hence the diameter = 2r = (0.3 2) cm = 0.6 cm

A

hemispherical bowl of internal radius 9 cm is full of water. Its contents are

emptied into a cylindrical vessel of internal radius 6 cm. Find the height of

water in the cylindrical vessel.

The

rain water from a roof of 44 m x 20 m drains into a cylindrical tank having

diameter of base 4 m and height 3.5 m. If the tank is just full, find the

rainfall in cm.

A

solid right circular cone of height 60 cm and radius 30 cm is dropped in a

right circular cylinder full of water, of height 180 cm and radius 60 cm.

Find the volume of water left in the cylinder, in cubic metres.

Water

is flowing through a cylindrical pipe of internal diameter 2 cm, into a

cylindrical tank of base radius 40 cm, at the rate of 0.4 m per second.

Determine the rise in level of water in the tank in half an hour.

Water

is flowing at the rate of 6 km/hr through a pipe of diameter 14 cm into a

rectangular tank which is 60 m long and 22 m wide. Determine the time in

which the level of water in the tank will rise by 7 cm.

Water

in a canal, 6 m wide and 1.5 m deep, is flowing at a speed of 4 km/hr. How

much area will it irrigate in 10 minutes if 8 cm of standing water is needed

for irrigation?

A farmer connects a pipe of internal diameter 25 cm from a canal into a cylindrical tank in his field which is 12 m in diameter and 2.5 m deep. If water flows through the pipe at the rate of 3.6 km/h, in how much time will the tank be filled? Also, find the cost of water if the canal department charges at the rate ofRs. 0.07. use

Height of cylindrical tank = 2.5 m

Its diameter = 12 m, Radius = 6 m

Volume of tank =

Water is flowing at the rate of 3.6 km/ hr = 3600 m/hr

Diameter of pipe = 25 cm, radius = 0.125 m

Volume of water flowing per hour

Water

running in a cylindrical pipe of inner diameter 7 cm, is collected in a

container at the rate of 192.5 litres per minute. Find the rate of flow of

water in the pipe in km/hr.

150

spherical marbles, each of diameter 14 cm, are dropped in a cylindrical

vessel of diameter 7 cm containing some water, which are completely immersed

in water. Find the rise in the level of water in the vessel.

Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm, containing some water. Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6 cm.

Let the number of marbles be n

n volume of marble = volume of rising water in beaker

In

a village, a well with 10 m inside diameter, is dug 14 m deep. Earth taken

out of it is spread all around to a width of 5 m to form an embankment. Find

the height of the embankment. What value of the villagers is reflected here?

In

a corner of a rectangular field with dimensions 35 m x 22 m, a well with 14 m

inside diameter is dug 8 m deep. The earth dug out is spread evenly over the

remaining part of the field. Find the rise in the level of the field.

A

copper wire of diameter 6 mm is evenly wrapped on a cylinder of length 18 cm

and diameter 49 cm to cover its whole surface. Find the length and the volume

of the wire. If the density of copper be 8.8 g per cu-cm, find the weight of

the wire.

A

right triangle whose sides are 15 cm and 20 cm (other than hypotenuse), is

made to revolve about its hypotenuse. Find the volume and surface area of the

double cone so formed. (Choose value of it as found appropriate)

## Chapter 19 – Volume and Surface Areas of Solids Exercise Ex. 19C

A

drinking glass is in the shape of a frustum of a cone of height 14 cm. The

diameters of its two circular ends are 16 cm and 12 cm. Find the capacity of

the glass.

The

radii of the circular ends of a solid frustum of a cone are 18 cm and 12 cm

and its height is 8 cm. Find its total surface area. [Use π

= 3.14.]

A

metallic bucket, open at the top, of height 24 cm is in the form of the

frustum of a cone, the radii of whose lower and upper circular ends are 7 cm

and 14 cm respectively. Find

(i) the volume of water which can

completely fill the bucket;

(ii) the

area of the metal sheet used to make the bucket.

A

container, open at the top, is in the form of a frustum of a cone of height

24 cm with radii of its lower and upper circular ends as 8 cm and 20 cm

respectively. Find the cost of milk which can completely fill the container

at the rate of Rs. 21 per litre.

A

container, open at the top and made up of metal sheet, is in the form of a

frustum of a cone of height 16 cm with diameters of its lower and upper ends

as 16 cm and 40 cm respectively. Find the cost of metal sheet used to make

the container, if it costs Rs. 10 per 100 cm^{2}.

The radii of the circular ends of a solid frustum of a cone are 33cm and 27 cm, and its slant height is 10 cm. Find its capacity and total surface area. Take .

Here R = 33 cm, r = 27 cm and l = 10 cm

Capacity of the frustum

Total surface area =

A bucket is in the form of a frustum of a cone. Its depth is 15 cm and the diameters of the top and the bottom are 56 cm and 42 cm, respectively. Find how many litres of water can the bucket hold. Take

Height = 15 cm, R = and

Capacity of the bucket =

Quantity of water in bucket = 28.49 litres

A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the bucket if the cost of metal sheet used is Rs. 15 per . Use

R = 20 cm, r = 8 cm and h = 16 cm

Total surface area of container =

Cost of metal sheet used =

A bucket made up of a metal sheet is in the form of frustum of a cone. Its depth is 24 cm and the diameters of the top and bottom are 30 cm and 10cm respectively. Find the cost of milk which can completely fill the bucket at the rate of Rs. 20 per litre and the cost of metal sheet used if it costs Rs. 10 per 100

R = 15 cm, r = 5 cm and h = 24 cm

(i)Volume of bucket =

Cost of milk = Rs. (8.164 20) = Rs. 163.28

(ii)Total surface area of the bucket

Cost of sheet =

A

container in the shape of a frustum of a cone having diameters of its two

circular faces as 35 cm and 30 cm and vertical height 14 cm, is completely

filled with oil. If each cm’ of oil has mass 1.2 g, then find the cost of oil

in the container if it costs Rs.40 per kg.

A

bucket is in the form of a frustum of a cone and it can hold 28.49 litres of

water. If the radii of its circular ends are 28 cm and 21 cm, find the height

of the bucket.

The

radii of the circular ends of a bucket of height 15 cm are 14 cm and r cm

< 14). If the volume of bucket is 5390 cm^{3}, find the value of

r.

The

radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm

and its slant height is 10 cm. Find its total surface area. [Use π

= 3.14.]

A tent is made in the form of a frustum of a cone surmounted by another cone. The diameters of the base and the top of the frustum are 20 m and 6 m respectively, and the height is 24 m. If the height of the tent is 28 m and the radius of the conical part is equal to the radius of the top of the frustum, find the quantity of canvas required. Take

R = 10cm, r = 3 m and h = 24 m

Let l be the slant height of the frustum, then

Quantity of canvas = (Lateral surface area of the frustum)

+ (lateral surface area of the cone)

A tent consists of a frustum of a cone, surmounted by a cone. If the diameters of the upper and lower circular ends of the frustum be 14 m and 26 m respectively, the height of the frustum be 8 m and the slant height of the surmounted conical portion be 12mm, find the area of the canvas required to make the tent. (Assume that the radii of the upper circular ends of the frustum and the base of the surmounted conical portion are equal.)

ABCD is the frustum in which upper and lower radii are EB = 7 m and FD = 13 m

Height of frustum= 8 m

Slant height of frustum

Radius of the cone = EB = 7 m

Slant height of cone = 12 m

Surface area of canvas required

The

perimeters of the two circular ends of a frustum of a cone are 48 cm and 36

cm. If the height of the frustum is 11 cm, find its volume and curved surface

area.

A solid cone of base radius 10 cm is cut into two parts through the midpoint of its height, by a plane parallel to its base. Find the ratio of the volumes of the two parts of the cone.

A

fez, the cap used by the Turks, is shaped like the frustum of a cone. If its

radius on the open side is 10 cm, radius at the upper base is 4 cm and its

slant height is 15 cm, find the area of material used for making it.

An

oil funnel made of tin sheet consists of a 10 cm long cylindrical portion

attached to a frustum of a cone. If the total height is 22 cm, diameter of

the cylindrical portion is 8 cm and the diameter of the top of the funnel is

18 cm, find the area of the tin sheet requited to make the funnel.

## Chapter 19 – Volume and Surface Areas of Solids Exercise Ex. 19D

A

river 1.5 m deep and 36 m wide is flowing at the rate of 3.5 km/hr. Find the

amount of water (in cubic metres) that runs into the sea per minute.

The

volume of a cube is 729 cm^{3}. Find its surface area.

How

many cubes of 10 cm edge can be put in a cubical box of 1 m edge?

Three

cubes of iron whose edges are 6 cm, 8 cm and 10 cm respectively are melted

and formed into a single cube. Find the edge of the new cube formed.

Five

identical cubes, each of edge 5 cm, are placed adjacent to each other. Find

the volume of the resulting cuboid.

The

volumes of two cubes are in the ratio 8 : 27. Find the ratio of their surface

areas.

The

ratio between the radius of the base and the height of a cylinder is 2 : 3.

If the volume of the cylinder is 12936 cm^{3}, find the radius of the

base of the cylinder.

The

radii of two cylinders are in the ratio of 2 : 3 and their heights are in the

ratio of 5 : 3. Find the ratio of their volumes.

66

cubic cm of silver is drawn into a wire 1 mm in diameter. Calculate the

length of the wire in metres.

If

the area of the base of a right circular cone is 3850 cm^{2} and its

height is 84 cm, find the slant height of the cone.

A

cylinder with base radius 8 cm and height 2 cm is melted to form a cone of

height 6 cm. Calculate the radius of the base of the cone.

A

right cylindrical vessel is full of water. How many right cones having the

same radius and height as those of the right cylinder will be needed to store

that water?

The

volume of a sphere is 4851 cm^{3}. Find its curved surface area.

The

curved surface area of a sphere is 5544 cm^{3}. Find its volume.

The

surface areas of two spheres are in the ratio of 4 : 25. Find the ratio of

their volumes.

A

solid metallic sphere of radius 8 cm is melted and recast into spherical

balls each of radius 2 cm. Find the number of spherical balls obtained.

How

many lead shots each 3 mm in diameter can be made from a cuboid of dimensions

9 cm x 11 cm x 12 cm?

A

metallic cone of radius 12 cm and height 24 cm is melted and made into

spheres of radius 2 cm each. How many spheres are formed?

A

hemisphere of lead of radius 6 cm is cast into a right circular cone of

height 75 cm. Find the radius of the base of the cone.

A

copper sphere of diameter 18 cm is drawn into a wire of diameter 4 mm. Find

the length of the wire.

The

radii of the circular ends of a frustum of height 6 cm are 14 cm and 6 cm

respectively. Find the slant height of the frustum.

Find

the ratio of the volume of a cube to that of a sphere which will fit inside

it.

Find

the ratio of the volumes of a cylinder, a cone and a sphere, if each has the

same diameter and same height?

Two

cubes each of volume 125 cm^{3} are joined end to end to form a

solid. Find the surface area of the resulting cuboid.

Three

metallic cubes whose edges are 3 cm, 4 cm and 5 cm, are melted and recast

into a single large cube. Find the edge of the new cube formed.

A

solid metallic sphere of diameter 8 cm is melted and drawn into a cylindrical

wire of uniform width. If the length of the wire is 12 m, find its width.

A

5-m-wide cloth is used to make a conical tent of base diameter 14 m and

height 24 m. Find the cost of cloth used, at the rate of Rs. 25 per metre.

A

wooden toy was made by scooping out a hemisphere of same radius from each end

of a solid cylinder. If the height of the cylinder is 10 cm and its base is

of radius 3.5 cm, find the volume of wood in the toy.

A

hollow sphere of external and internal diameters 8 cm and 4 cm respectively

is melted into a solid cone of base diameter 8 cm. Find the height of the

cone.

A

bucket of height 24 cm is in the form of frustum of a cone whose circular

ends are of diameter 28 cm and 42 cm. Find the cost of milk at the rate of Rs.

30 per litre, which the bucket can hold.

The

interior of a building is in the form of a right circular cylinder of

diameter 4.2 m and height 4 m surmounted by a cone of same diameter. The

height of the cone is 2.8 m. Find the outer surface area of the building.

A

metallic solid right circular cone is of height 84 cm and the radius of its

base is 21 cm. It is melted and recast into a solid sphere. Find the diameter

of the sphere.

A

toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same

radius. The total height of the toy is 15.5 cm. Find the total surface area

of the toy.

If

the radii of the circular ends of a bucket 28 cm high, are 28 cm and 7 cm,

find its capacity and total surface area.

A

bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm^{3}

of water. The radii of the top and bottom circular ends are 20 cm and 12 cm

respectively. Find the height of the bucket. (Useπ

= 3.14.)

The

radii of its lower and upper circular ends are 8 cm and 20 cm respectively.

Find the cost of metal sheet used in making the container at the rate of Rs.1.40 per cm^{2}.

A

cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of

water. A solid cone of base diameter 7 cm and height 6 cm is completely

immersed in water. Find the volume of water

(i) displaced out of the cylinder

(ii) left

in the cylinder.

## Chapter 19 – Volume and Surface Areas of Solids Exercise FA

Find the number of solid

spheres, each of diameter 6 cm, that could be moulded

to form a solid metallic cylinder of height 45 cm and diameter 4 cm.

Two right circular

cylinders of equal volumes have their heights in the ratio 1: 2. What is the

ratio of their radii?

A circus tent is

cylindrical to a height of 4 m and conical above it. If its diameter is 105 m

and its slant height is 40 m, find the total area of the canvas required.

The radii of the top and

bottom of a bucket of slant height 45 cm are 28 cm and 7 cm respectively.

Find the curved surface area of the bucket.

A solid metal cone with

radius of base 12 cm and height 24 cm is melted to form solid spherical balls

of diameter 6 cm each. Find the number of balls formed.

A hemispherical bowl of

internal diameter 30 cm is full of a liquid. This liquid is filled into

cylindrical-shaped bottles each of diameter 5 cm and height 6 cm. How many

bottles are required?

A solid metallic sphere

of diameter 21 cm is melted and recast into Milan cones, each of diameter 3.5

cm and height 3 cm. Find the number of cones so formed.

The diameter of a sphere

is 42 cm. it is melted and drawn into a cylindrical wire of diameter 2.8 cm.

Find the length of the wire.

A drinking glass is in

the shape of frustum of a cone of height 21 cm with 6 cm and 4 cm as the

diameters of its two circular ends. Find the capacity of the glass.

Two cubes, each of volume

64 cm^{3}, are joined end to end. Find the total surface area of the

resulting cuboid.

A toy is in the form of

a cone mounted on a hemisphere of common base radius 7 cm. The total height

of the toy is 31 cm. Find the total surface area of the toy.

A hemispherical bowl of

internal radius 9 cm is full of water. This water is to be filled in

cylindrical bottles of diameter 3 cm and height 4 cm. Find the number of

bottles needed to fill the whole water of the bowl.

The slant height of the

frustum of a cone is 4 cm and the perimeters (i.e., circumferences) of its

circular ends are 18 cm and 6 cm. Find the curved surface area of the

frustum.

A solid is composed of a

cylinder with hemispherical ends. If the whole length of the solid is 104 cm

and the radius of each hemispherical end is 7 cm, find the surface area of

the solid.

From a solid cylinder

whose height is 15 cm and diameter 16 cm, a conical cavity of the same height

and same “diameter is hollowed out. Find the total surface area of the

remaining solid. (Use 𝜋 = 3.14.)

A solid rectangular

block of dimensions 4.4 m, 2.6 in and 1 m is cast into a hollow cylindrical

pipe of internal radius 30 cm and thickness 5 cm. Find the length of the

pipe.

An open metal bucket is

in the shape of a frustum of a cone, mounted on a hollow cylindrical base

made of the same metallic sheet. The diameters of the two circular ends of

the bucket are 45 cm and 25 cm, the total vertical height of the bucket is 40

cm and that of the cylindrical base is 6 cm. Find the area of the metallic

sheet used to make the bucket. Also, find the volume of water the bucket can

hold, in litres.

A fanner connects a pipe

of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m

in diameter and 2 m deep. If the water flows through the pipe at the rate of

4 km/hr, in how much time will the tank be filled completely?

## Chapter 19 – Volume and Surface Areas of Solids Exercise MCQ

Choose the correct

answer in each of the following:

A cylindrical pencil sharpened at one

edge is the

combination of

(a) a cylinder and

a cone

(b) a cylinder and

frustum of a cone

(c) a cylinder and

a hemisphere

(d) two cylinders

Correct

option: (a)

A cylindrical pencil sharpened at one

edge is the combination of a cylinder and a cone. Observe the figure, the

lower portion is a cylinder and the upper tapering portion is a cone.

A shuttlecock used for playing badminton

is the combination of

(a) cylinder and a

hemisphere

(b) frustum of a

cone and a hemisphere

(c) a cone and a

hemisphere

(d) a cylinder and

a sphere

Correct option: (b)

A shuttlecock used for

playing badminton is the combination of a frustum of a cone and a hemisphere,

the lower portion being the hemisphere and the portion above that being the

frustum of the cone.

A funnel is the combination of

(a) a cylinder and

a cone

(b) a cylinder and

a hemisphere

(c) a cylinder and

frustum of a cone

(d) a cone and a

hemisphere

Correct

option: (c)

A funnel is the combination of a cylinder

and frustum of a cone. The lower portion is cylindrical and the upper portion

is a frustum of a cone.

A surahi is a

combination of

(a) a sphere and a

cylinder

(b) a hemisphere

and a cylinder

(c) a cylinder and

a cone

(d) two hemispheres

Surahi

Correct

option: (a)

A surahi is a

combination of a sphere and a cylinder, the lower portion is the sphere and

the upper portion is the cylinder.

The shape of a glass (tumbler) is usually

in the form of

(a) a cylinder

(b) frustum of a

cone

(c) a cone

(d) a sphere Glass

Correct

option: (b)

The shape of a glass (tumbler)

is usually in the form of a frustum of a cone.

The shape of a gill in the gilli-danda game is a combination of

(a) a cone and a

cylinder

(b) two cylinders Gilli

(c) two cones and a

cylinder

(d) two cylinders

and a cone

Correct

option: (c)

The shape of a gill in the gilli-danda game is a combination of two cones and a

cylinder. The cones at either ends with the cylinder in the middle.

A plumbline (sahul) is the combination of

(a) a hemisphere

and a cone

(b) a cylinder and

a cone

(c) a cylinder and

frustum of a cone

(d) a cylinder and

a sphere Plumbline

Correct

option: (a)

A plumbline (sahul) is the combination of a hemisphere and a cone, the

hemisphere being on top and the lower portion being the cone.

A cone is cut by a plane

parallel to its base and the upper part is removed. The part that is left

over is called

(a) a cone

(b) a sphere

(c) a cylinder

(d) frustum of a

cone

Correct option: (d)

A cone is cut by a plane parallel to its

base and the upper part is removed. The part that is left over is called the

frustum of a cone.

During conversion of a solid from one

shape to another, the volume of the new shape will

(a) decrease

(b) increase

(c) remain

unaltered

(d) be doubled

Correct

option: (c)

During conversion of a solid from one

shape to another, the volume of the new shape will remain altered.

In a right circular cone, the cross

section made by a plane parallel to the base is a

(a) sphere

(b) hemisphere

(c) circle

(d) a semicircle

Correct option: (c)

In a right circular cone, the cross

section made by a plane parallel to the base is a circle.

A solid piece of iron in the form of a cuboid of dimensions (cccm) is moulded to form a solid sphere. The radius of the sphere

is

(a) 19 cm

(b) 21 cm

(c) 23 cm

(d) 25 cm

The radius (in cm) of the largest right

circular cone that can be cut out from a cube of edge 4.2 cm is

(a) 2.1

(b) 4.2

(c) 8.4

(d) 1.05

A metallic solid sphere of radius 9 cm is

melted to form a solid cylinder of radius 9 cm. The height of the cylinder is

(a) 12 cm

(b) 18 cm

(c) 36 cm

(d) 96 cm

A rectangular sheet of paper 40 cm × 22 cm,

is rolled to form a hollow cylinder of height 40 cm. The radius of the

cylinder (in cm) is,

The number of solid spheres, each of

diameter 6 cm, that can be made by melting a solid metal cylinder of height

45 cm and diameter 4 cm, is

(a) 2

(b) 4

(c) 5

(d) 6

The surface areas of two spheres are in

the ratio 16 : 9. The ratio of their volumes is

(a) 64 : 27

(b) 16:9

(c) 4 :3

(d) 16^{3}

: 9^{3}

If the surface area of a sphere is 616 cm^{2},

its diameter (in cm) is

(a) 7

(b) 14

(c) 28

(d) 56

If the radius of a sphere becomes 3 times

then its volume will become

(a) 3 times

(b) 6 times

(c) 9 times

(d) 27 times

If the height of a bucket in the shape of

frustum of a cone is 16 cm and the diameters of its two circular ends are 40

cm and 16 cm then its slant height is

A sphere of diameter 18 cm is dropped

into a cylindrical vessel of diameter 36 cm, partly

filled with water. If the sphere is completely submerged then the water level

rises by

(a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

A solid right circular cone is cut into

two parts at the middle of its height by a plane parallel to its base. The

ratio of the volume of the smaller cone to the whole cone is

(a) 1 : 2

(b) 1 : 4

(c) 1 : 6

(d) 1 : 8

The radii of the circular ends of a

bucket of height 40 cm are 24 cm and 15 cm. The slant height (in cm) of the

bucket is

(a) 41

(b) 43

(c) 49

(d) 51

A solid is hemispherical at the bottom

and conical (of same radius) above it. If the surface areas of the two parts

are equal then the ratio of its radius and the slant height of the conical

part is

(a) 1 : 2

(b) 2 : 1

(c) 1 : 4

(d) 4 : 1

If the radius of the base of a right

circular cylinder is halved, keeping the height the same, then the ratio of

the volume of the cylinder thus obtained to the volume of original cylinder

is

(a) 1 : 2

(b) 2 : 1

(c) 1 : 4

(d) 4: 1

A cubical ice-cream brick of edge 22 cm

is to be distributed among some children by filling ice-cream cones of radius

2 cm and height 7 cm up to its brim. How many children will get the ice-cream

cones?

(a) 163

(b) 263

(c) 363

(d) 463

(a) 11000

(b) 11100

(c) 11200

(d) 11300

Twelve solid spheres of the same size are

made by melting a solid metallic cylinder of base diameter 2 cm and height 16

cm. The diameter of each sphere is

(a) 2 cm

(b) 3 cm

(c) 4 cm

(d) 6 cm

The diameters of two circular ends of a

bucket are 44 cm and 24 cm, and the height of the bucket is 35 cm. The

capacity of the bucket is

(a) 31.7 litres

(b) 32.7 litres

(c) 33.7 litres

(d) 34.7 litres

The slant height of a bucket is 45 cm and

the radii of its top and bottom are 28 cm and 7 cm respectively. The curved

surface area of the bucket is

(a) 4953 cm^{2}

(b) 4952 cm^{2}

(c) 4951 cm^{2 }

(d) 4950 cm^{2}

The volumes of two spheres are in the

ratio 64:27. The ratio of their surface area is

(a) 9:16

(b) 16:9

(c) 3:4

(d) 4:3

(a) 142296

(b) 142396

(c) 142496

(d) 142596

A metallic spherical shell of internal

and external diameters 4 cm and 8 cm respectively, is melted and recast into

the form of a cone of base diameter 8 cm. The height of the cone is

(a) 12 cm

(b) 14 cm

(c) 15 cm

(d) 8 cm

A medicine capsule is in the shape of a

cylinder of diameter 0.5 cm with two hemispheres stuck to each of its ends.

The length of the entire capsule is 2 cm. The capacity of the capsule is

(a) 0.33 cm^{3}

(b) 0.34 cm^{3}

(c) 0.35 cm^{3}

(d) 0.36 cm^{3}

The length of the longest pole that can

be kept in a room (12 m × 9 m × 8 m) is

(a) 29 m

(b) 21 m

(c) 19 m

(d) 17 m

The

volume of a cube is 2744 cm^{3}. Its surface area is

(a) 196 cm^{2}

(b) 1176 cm^{2}

(c) 784 cm^{2}

(d) 588 cm^{2}

The

total surface area of a cube is 864 cm^{2}. Its volume is

(a) 3456 cm^{3}

(b) 432 cm^{3 }

(c) 1728 cm^{3}

(d) 3456 cm^{3}

How many bricks each measuring (25 cm ×

11.25 cm × 6 cm) will be required to construct a wall (8 m × 6 m × 22.5 cm)?

(a) 8000

(b) 6400

(c) 4800

(d) 7200

The area of the base of a rectangular

tank is 6500 cm^{2} and the volume of water contained in it is 2.6 m^{3}.

The depth of water in the tank is.

(a) 3.5 m

(b) 4 m

(c) 5 m

(d) 8 m

The volume of a wall, 5 times as high as

it is broad and 8 times as long as it is high, is 12.8 m^{3}. The

breadth of the wall is

(a) 30 cm

(b) 40 cm

(c) 22.5 cm

(d) 25 cm

If the areas of three adjacent faces of a

cuboid are x, y, z respectively then the volume of

the cuboid is

(a) 361 cm^{2}

(b) 125 cm^{2}

(c) 236 cm^{2}

(d) 486 cm^{2}

If each edge of a cube is increased by

50%, the percentage increase in the surface area is

(a) 50%

(b) 75%

(c) 100%

(d) 125%

How many bags of grain can be stored in a

cuboidal granary (8 m× 6m × 3 m), if each bag

occupies a space of 0.64 m^{3}?

(a) 8256

(b) 90

(c) 212

(d) 225

A cube of side 6 cm is cut into a number

of cubes each of side 2 cm. The number of cubes formed is

(a) 6

(b) 9

(c) 12

(d) 27

In a shower, 5 cm of rain falls. The

volume of the water that falls on 2 hectares of ground, is

(a) 100 m^{3}

(b) 10 m^{3}

(c) 1000 m^{3}

(d) 10000 m^{3}

Two cubes have their volumes in the ratio

1: 27. The ratio of their surface areas is

(a) 1 : 3

(b) 1 : 8

(c) 1 : 9

(d) 1 : 18

The diameter of the base of a cylinder is

4 cm and its height is 14 cm. The volume of the cylinder is

(a) 176 cm^{3}

(b) 196 cm^{3}

(c) 276 cm^{3}

(d) 352 cm^{3}

The diameter of a cylinder is 28 cm and

its height is 20 cm. The total surface area of the cylinder is

(a) 2993 cm^{2}

(b) 2992 cm^{2}

(c) 2292 cm^{2}

(d) 2229 cm^{2}

The height of a cylinder is 14 cm and its

curved surface area is 264 cm^{2}. The volume of the cylinder is

(a) 308 cm^{3}

(b) 396 cm^{3}

(c) 1232 cm^{3}

(d) 1848 cm^{3}

The curved surface area of a cylinder is

1760 cm^{2} and its base radius is 14 cm. The height of the cylinder

is

(a) 10 cm

(b) 15 cm

(c) 20 cm

(d) 40 cm

The ratio of the total surface area to

the lateral surface area of a cylinder with base radius 80 cm and height 20

cm is

(a) 2 :1

(b) 3:1

(c) 4:1

(d) 5:1

The curved surface area of a cylindrical

pillar is 264 m^{2} and its volume is 924 m^{3}. The height

of the pillar is

(a) 4 m

(b) 5 m

(c) 6 m

(d) 7 m

The ratio between the radius of the base

and the height of the cylinder is 2 : 3. If its

volume is 1617 cm^{3}, the total surface area of the cylinder is

(a) 308 cm^{2}

(b) 462 cm^{2}

(c) 540 cm^{2}

(d) 770 cm^{2}

The radii of two cylinders are in the

ratio 2:3 and their heights are in the ratio 5 : 3.

The ratio of their volumes is

(a) 27 : 20

(b) 20 : 27

(c) 4 :9

(d) 9 : 4

Two circular cylinders of equal volume have

their heights in the ratio 1:2. The ratio of their radii is

The radius of the base of a cone is 5 cm

and its height is 12 cm. Its curved surface area is

The diameter of the base of a cone is 42

cm and its volume is 12936 cm^{3}. Its height is

(a) 28 cm

(b) 21 cm

(c) 35 cm

(d) 14 cm

The area of the base of a right circular

cone is 154 cm^{2 }and its height is 14 cm. Its curved surface area

is

On increasing each of the radius of the

base and the height of a cone by 20% its volume will be increased by

(a) 20%

(b) 40%

(c) 60%

(d) 72.8%

The radii of the base of a cylinder and a

cone are in the ratio 3:4. If they have their heights in the ratio 2 : 3, the

ratio between their volumes is

(a) 9 :8

(b) 3:4

(c) 8 :9

(d) 4 : 3

A metallic cylinder of radius 8 cm and

height 2 cm is melted and converted into a right circular cone of height 6

cm. The radius of the base of this cone is

(a) 4 cm

(b) 5 cm

(c) 6 cm

(d) 8 cm

The height of a conical tent is 14 m and

its floor area is 346.5 m^{2}. How much canvas, 1.1 m wide, will be

required for it?

(a) 490 m

(b) 525 m

(c) 665 m

(d) 860 m

The

diameter of a sphere is 14 cm. Its volume is

The ratio between the volumes of two

spheres is 8: 27. What is the ratio between their surface areas?

(a) 2:3

(b) 4:5

(c) 5:6

(d) 4: 9

A hollow metallic sphere with external

diameter 8 cm and internal diameter 4 cm is melted and moulded

into a cone having base radius 8 cm. The height of the cone is

(a) 12 cm

(b) 14 cm

(c) 15 cm

(d) 18 cm

A metallic cone having base radius 2.1 cm

and height 8.4 cm is melted and moulded into a

sphere. The radius of the sphere is

(a) 2.1 cm

(b) 1.05 cm

(c) 1.5 cm

(d) 2 cm

The volume of a hemisphere is 19404 cm^{3}.

The total surface area of the hemisphere is

(a) 4158 cm^{2}

(b) 16632 cm^{2}

(c) 8316 cm^{2}

(d) 3696 cm^{2}

Correct option: (a)

The surface area of a sphere is 154 cm^{2}.

The volume of the sphere is all

Correct option: (a)

The

total surface area of a hemisphere of radius 7 cm is

(588

𝜋) cm^{2}

(392

𝜋) cm^{2}

(147

𝜋) cm^{2}

(598

𝜋) cm^{2}

The circular ends of a bucket are of

radii 35 cm and 14 cm and the height of the bucket is 40 cm. Its volume is

(a) 60060 cm^{3}

(b) 80080 cm^{3}

(c) 70040 cm^{3}

(d) 80160 cm^{3}

If the radii of the ends of a bucket are

5 cm and 15 cm and it is 24 cm high then its surface area is

(a) 1815.3 cm^{2}

(b) 1711.3 cm^{2}

(c) 2025.3 cm^{2 }

(d) 2360 cm^{2}

A circus tent is cylindrical to a height

of 4 m and conical above it. If its diameter is 105 m and its slant height is

40 m, the total area of canvas required is

(a) 1760 m^{2}

(b) 2640 m^{2}

(c) 3960 m^{2}

(d) 7920 m^{2}

Match

the following columns:

Column I | Column II |

A solid metallic sphere of | (p) 18 |

A 20-in-deep well with diameter | (q) 8 |

A sphere of radius 6 cm is melted | (r) 16 : 9 |

The volumes of two spheres are in | (s) 5 |

The

correct answer is

(a)-….., (b)- ….. , (c)- ….., (d)- ……

Match

the following columns:

Column I | Column II |

The radii of the circular ends | (p) 2418 π |

The radii of the circular ends | (q) 22000 |

The radii of the circular ends of | (r) 12 |

Three solid metallic spheres of | (s) 17 |

The correct answer is

(a)-….., (b)-

….. , (c)- ….., (d)- ……

Assertional– and-Resons type

Each

question consists of two statements, namely,

Assertion

(A) and Reason (R). For selecting the correct

answer,

use the following code:

(a) Both Assertion

(A) and Reason (R) are true and Reason (R) is a correct explanation of

Assertion (A).

(b) Both Assertion

(A) and Reason (R) are true but Reason (R) is not a correct explanation of

Assertion (A).

(c) Assertion (A)

is true and Reason (R) is false.

(d) Assertion (A)

is false and Reason (R) is true.

Assertion (A) | Reason (R) |

If the radii of the circular | If the radii of the circular |

The correct answer is (a)/(b)/(c)

/(d) .

Assertional– and-Resons type

Each

question consists of two statements, namely,

Assertion

(A) and Reason (R). For selecting the correct

answer,

use the following code:

(a) Both Assertion

(A) and Reason (R) are true and Reason (R) is a correct explanation of

Assertion (A).

(b) Both Assertion

(A) and Reason (R) are true but Reason (R) is not a correct explanation of

Assertion (A).

(c) Assertion (A)

is true and Reason (R) is false.

(d) Assertion (A)

is false and Reason (R) is true.

Assertion (A) | Reason (R) |

A hemisphere of radius 7 cm is | The total surface area |

The correct answer is (a)/(b)/(c)

/(d) .

Assertional– and-Resons type

Each

question consists of two statements, namely,

Assertion

(A) and Reason (R). For selecting the correct

answer,

use the following code:

(a) Both Assertion

(A) and Reason (R) are true and Reason (R) is a correct explanation of

Assertion (A).

(b) Both Assertion

(A) and Reason (R) are true but Reason (R) is not a correct explanation of

Assertion (A).

(c) Assertion (A)

is true and Reason (R) is false.

(d) Assertion (A)

is false and Reason (R) is true.

Assertion (A) | Reason (R) |

The number of coins 1.75 cm in | Volume of a cylinder of base V = (πr And, area of a cuboid = (l × b × h) cubic units. |

The

correct answer is (a)/(b)/(c) /(d) .

Assertional– and-Resons type

Each

question consists of two statements, namely,

Assertion

(A) and Reason (R). For selecting the correct

answer,

use the following code:

(A) and Reason (R) are true and Reason (R) is a correct explanation of

Assertion (A).

(A) and Reason (R) are true but Reason (R) is not a correct explanation of

Assertion (A).

(c) Assertion (A)

is true and Reason (R) is false.

(d) Assertion (A)

is false and Reason (R) is true.

Assertion (A) | Reason (R) |

If the volumes of two spheres |

The

correct answer is (a)/(b)/(c) /(d) .

Assertional– and-Resons type

Each

question consists of two statements, namely,

Assertion

(A) and Reason (R). For selecting the correct

answer,

use the following code:

(A) and Reason (R) are true and Reason (R) is a correct explanation of

Assertion (A).

(A) and Reason (R) are true but Reason (R) is not a correct explanation of

Assertion (A).

(c) Assertion (A)

is true and Reason (R) is false.

(d) Assertion (A)

is false and Reason (R) is true.

Assertion (A) | Reason (R) |

The curved surface area of a | Volume of a cone = πr |

The

correct answer is (a)/(b)/(c) /(d) .