# R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 15 - Volume and Surface Area of Solids

## Chapter 15 - Volume and Surface Area of Solids Exercise MCQ

The length, breadth and height of a cuboid are 15 cm, 12 cm and 4.5 cm respectively. Its volume is

- 243
cm
^{3} - 405
cm
^{3} - 810
cm
^{3} - 603
cm
^{3}

A cuboid is 12 cm long, 9 cm broad and 8 cm high. Its total surface area is

- 864
cm
^{2} - 552
cm
^{2} - 432
cm
^{2} - 276
cm
^{2}

The length breadth and height of a cuboid are 15m, 6m, and 5 dm respectively. The lateral surface area of the cuboid is

- 45
m
^{2} - 21
m
^{2} - 201
m
^{2} - 90
m
^{2}

A beam 9 m long, 40 cm wide and 20 cm high is made up of iron which weighs 50 kg per cubic metre. The weight of the beam is

- 27 kg
- 48 kg
- 36 kg
- 56 kg

The length of the longest rod that can be placed in a room of dimensions (10 m × 10 m × 5 m) is

- 15 m
- 16 m
- 12 m

What is the maximum length of a pencil that can be placed in a rectangular box of dimensions (8 cm × 6 cm × 5 cm)?

- 8 cm
- 9.5 cm
- 19 cm
- 11.2 cm

The number of planks of dimensions (4 m × 5 m × 2 m) that can be stored in a pit which is 40 m long, 12 m wide and 16 m deep is

- 190
- 192
- 184
- 180

How many planks of dimensions (5 m × 25 cm × 10 cm) can be stored in a pit which is 20 m long, 6 m wide and 50 cm deep?

- 480
- 450
- 320
- 360

How many bricks will be required to construct a wall 8 m long, 6 m high and 22.5 cm thick if each brick measures (25 cm × 11.25 cm × 6 cm)?

- 4800
- 5600
- 6400
- 5200

How many persons can be
accommodated in a dining hall of dimensions (20 m × 15 m × 4.5 m),
assuming that each person requires 5 m^{3} of air?

- 250
- 270
- 320
- 300

A river 1.5 m deep and 30 m wide is flowing at the rate of 3 km per hour. The volume of water that runs into the sea per minute is

- 2000
m
^{3} - 2250
m
^{3} - 2500
m
^{3} - 2750
m
^{3}

The lateral surface area
of a cube is 256 m^{2}. The volume of the cube is

- 64
m
^{3} - 216
m
^{3} - 256
m
^{3} - 512
m
^{3}

The total surface area
of a cube is 96m^{2}. The volume of the cube is

- 8
cm
^{3} - 27cm
^{3} - 64cm
^{3} - 512
cm
^{3}

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

- 256
cm
^{2} - 384
cm
^{2} - 512
cm
^{2} - 64
cm
^{2}

The length of the longest rod that can fit in a cubical vessel of side 10 cm is

- 10 cm
- 20 cm

If the length of diagonal of a cube is cm, then its surface area is

- 192
cm
^{2} - 384
cm
^{2} - 512
cm
^{2} - 768
cm
^{2}

If each edge of a cube is increased by 50%, then the percentage increase in its surface area is

- 50%
- 75%
- 100%
- 125%

Three cubes of metal with edges 3 cm, 4 cm and 5 cm respectively are melted to form a single cube. The lateral surface area of the new cube formed is

- 72
cm
^{2} - 144
cm
^{2} - 128
cm
^{2} - 256
cm
^{2}

In a shower, 5 cm of rain falls, what is the volume of water that falls on 2 hectors of ground?

- 500
m
^{3} - 750
m
^{3} - 800
m
^{3} - 1000
m
^{3}

Two cubes have their volumes in the ratio 1:27. The ratio of their surface area is

- 1:3
- 1:8
- 1:9
- 1:18

If each side of a cube is doubled, then its volume

- is doubled
- becomes 4 times
- becomes 6 times
- becomes 8 times

The diameter of a base of a cylinder is 6 cm and its height is 14 cm. The volume of the cylinder is

a. 198 cm^{3}

b. 396 cm^{3}

c. 495 cm^{3}

d. 297 cm^{3}

The diameter of a cylinder is 28 cm and its height is 20 cm, then its curved surface area is

- 880
cm
^{2} - 1760
cm
^{2} - 3520
cm
^{2} - 2640
cm
^{2}

If the curved surface
area of a cylinder is 1760 cm^{2} and its base radius is 14 cm, then
its height is

- 10 cm
- 15 cm
- 20 cm
- 40 cm

The height of a cylinder is 14 cm and its curved surface area is 264 cm^{2}. The volume of the cylinder is

- 308 cm
^{2} - 396 cm
^{2} - 1232 cm
^{2} - 1848 cm
^{2}

The curved surface area
of the cylindrical pillar is 264 m^{2} and its volume is 924m^{3}.
The height of the pillar is

- 4 m
- 5 m
- 6 m
- 7 m

The radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3. The ratio of their surface area is

- 2:5
- 8:7
- 10:9
- 16:9

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

- 27:20
- 20:27
- 4:9
- 9:4

The ratio between the
radius of the base and height of a cylinder is 2:3. If its volume is 1617 cm^{3},
then its total surface area is

- 308
cm
^{2} - 462
cm
^{2} - 540
cm
^{2} - 770
cm
^{2}

Two circular cylinders of equal volume have their heights in the ratio 1:2. The ratio of their radii is

The ratio between the
curved surface area and the total surface area of a right circular cylinder
is 1:2. If the total surface area is 616 cm^{2}, then the volume of
the cylinder is

- 1078
cm
^{3} - 1232
cm
^{3} - 1848
cm
^{3} - 924
cm
^{3}

In a cylinder, if the radius is halved and the height is doubled, then the volume will be

- The same
- Doubled
- Halved
- Four times

The number of coins 1.5 cm in diameter and 0.2 cm thick to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm is

- 540
- 450
- 380
- 472

The radius of a wire is decreased to one-third. If volume remains the same, the length will become

- 2 times
- 3 times
- 6 times
- 9 times

The diameter of a roller, 1m long is 84 cm. If it takes 500 complete revolutions to level a playground, the area of the playground is

- 1440
m
^{2} - 1320
m
^{2} - 1260
m
^{2} - 1550
m
^{2}

2.2 dm^{3} of
lead is to be drawn into a cylindrical wire 0.50 cm in diameter. The length
of the wire is

- 110 m
- 112 m
- 98 m
- 124 m

The lateral surface area of a cylindrical is

The height of a cone is 24 cm and the diameter of its base is 14 cm. The curved surface area of the cone is

- 528
cm
^{2} - 550
cm
^{2} - 616
cm
^{2} - 704
cm
^{2}

The volume of a right circular cone of height is 12 cm and base radius 6 cm, is

- (12π) cm
^{3} - (36π) cm
^{3} - (72π) cm
^{3} - (144π) cm
^{3}

How much cloth 2.5 m wide will be required to make a conical tent having base radius 7 m and height 24 m?

- 120 m
- 180 m
- 220 m
- 550 m

The volume of a cone is
1570 cm^{3} and its height is 15 cm. What is the radius of the cone?
(Use π = 3.14)

- 10 cm
- 9 cm
- 12 cm
- 8.5 cm

The height of cone is 21 cm and its slant height is 28 cm. The volume of the cone is

- 7356 cm
^{3} - 7546 cm
^{3} - 7506 cm
^{3} - 7564 cm
^{3}

Correct option: (b)

The volume of a right
circular cone of height 24 cm is 1232 cm^{3}. Its curved surface area
is

- 1254
cm
^{2} - 704
cm
^{2} - 550
cm
^{2} - 462
cm
^{2}

If the volumes of two cones be in the ratio 1:4 and the radii of their bases be in the ratio 4:5, then the ratio of their heights is

- 1:5
- 5:4
- 25:16
- 25:64

If the height of a cone is doubled, then its volume is increased by

- 100%
- 200 %
- 300 %
- 400 %

The curved surface area of the cone is twice that of the other while the slant height of the latter is twice that of the former. The ratio of their radii is

- 2:1
- 4:1
- 8:1
- 1:1

The ratio of the volumes of a right circular cylinder and a right circular cone of the same base and same height will be

- 1:3
- 3:1
- 4:3
- 3:4

A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is

- 3:5
- 2:5
- 3:1
- 1:3

The radii of the bases of a cylinder and a cone are in the ratio 3:4 and their heights are in the ratio 2:3. Then their volumes are in the ratio

- 9:8
- 8:9
- 3:4
- 4:3

If the height and the radius of cone are doubled, the volume of the cone becomes

- 3 times
- 4 times
- 6 times
- 8 times

A solid metallic cylinder of base radius 3 cm and height 5 cm is melted to make n solid cones of height 1 cm and base radius 1 mm. The value of n is

- 450
- 1350
- 4500
- 13500

A conical tent is to accommodate 11
persons such that each person occupies 4 m^{2} of space on the
ground. They have 220m^{3} of air to breathe. The height of the cone
is

- 14m
- 15 m
- 16 m
- 20 m

The volume of a sphere of radius 2r is

The volume of a sphere of a radius 10.5 cm is

- 9702
cm
^{3} - 4851
cm
^{3} - 19404
cm
^{3} - 14553
cm
^{3}

The surface area of a sphere of radius 21 cm is

- 2772
cm
^{2} - 1386
cm
^{2} - 4158
cm
^{2} - 5544
cm
^{2}

The surface area of a sphere is 1386 cm^{2}.
Its volume is

- 1617
cm
^{3} - 3234
cm
^{3} - 4851
cm
^{3} - 9702
cm
^{3}

If the surface area of a sphere is (144 π) m^{2},
then its volume is

- (288
π) m
^{3} - (188
π) m
^{3} - (300
π) m
^{3} - (316
π) m
^{3}

The volume of a sphere is 38808 cm^{3}.
Its curved surface area is

- 5544
cm
^{2} - 8316
cm
^{2} - 4158
cm
^{2} - 1386
cm
^{2}

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

- 1:2
- 1:4
- 1:8
- 1:16

A solid metal ball of radius 8 cm is melted and cast into smaller balls, each of radius 2 cm, The number of such balls is

- 8
- 16
- 32
- 64

A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. The radius of the sphere is

- 4.2 cm
- 2.1 cm
- 2.4 cm
- 1.6 cm

A solid lead ball of radius 6 cm is melted and then drawn into a wire of diameter 0.2 cm. The length of wire is

- 272 m
- 288 m
- 292 m
- 296 m

A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. The number of such cones will be

- 21
- 63
- 126
- 130

How many lead shots, each 0.3 cm in diameter, can be made from a cuboid of dimensions 9 cm × 11 cm × 12 cm?

- 7200
- 8400
- 72000
- 84000

The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 2 mm. The length of the wire is

- 12 m
- 18 m
- 36 m
- 66 m

A sphere of diameter 12.6 cm is melted and cast into a right circular cone of height 25.2 cm. The radius of the base of the cone is

- 6.3 cm
- 2.1 cm
- 6 cm
- 4 cm

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

- 1 cm
- 1.5 cm
- 2.5 cm
- 0.5 cm

The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratio of the surface areas of the balloons in two cases is

- 1:4
- 1:3
- 2:3
- 1:2

The volumes of the two spheres are in the ratio 64:27 and the sum of their radii is 7 cm. The difference of their total surface areas is

- 38
cm
^{2} - 58
cm
^{2} - 78
cm
^{2} - 88
cm
^{2}

A hemispherical bowl of radius 9 cm contains a liquid. This liquid is to be filled into cylindrical small bottles of diameter 3 cm and height 4 cm. How many bottles will be needed to empty the bowl?

- 27
- 35
- 54
- 63

A cone and a hemisphere have equal bases and equal volumes. The ratio of their heights is

- 1:2
- 2:1
- 4:1

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is

- 1:2:3
- 2:1:3
- 2:3:1
- 3:2:1

If the volumes and the surface area of sphere are numerically the same, then its radius is

- 1 units
- 2 units
- 3 units
- 4 units

## Chapter 15 - Volume and Surface Area of Solids Exercise Ex. 15C

Find the curved surface area of a cone with base radius 5.25 cm and slant height 10 cm.

Radius of a cone, r = 5.25 cm

Slant height of a cone, l = 10 cm

Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.

Radius of a cone, r = 12 m

Slant height of a cone, l = 21 cm

A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.

Radius of a conical cap, r = 7 cm

Height of a conical cap, h = 24 cm

Thus, 5500 cm^{2}
sheet will be required to make 10 caps.

The curved surface area of a cone
is 308 cm^{2} and its slant height is 14 cm. Find the radius of the
base and total surface area of the cone.

Let r be the radius of a cone.

Slant height of a cone, l = 14 cm

Curved
surface area of a cone = 308 cm^{2}

The slant height and base diameter
of a conical tomb are 25 m and 14 m respectively. Find the cost of
whitewashing its curved surface at the rate of Rs.12 per m^{2}.

Radius of a cone, r = 7 m

Slant height of a cone, l = 25 m

Cost
of whitewashing = Rs. 12 per m^{2}

⇒ Cost of
whitewashing 550 m^{2} area = Rs. (12 ×
550) = Rs. 6600

A conical tent is 10 m high and
radius of its base is 24 m. Find the slant height of the tent. If the cost of
1 m^{2} canvas is Rs.70, find the cost of canvas required to make the tent.

Radius of a conical tent, r = 24 m

Height of a conical tent, h = 10 m

A bus stop is barricaded from the
remaining part of the road by using 50 hollow cones made of recycled
cardboard. Each one has a base diameter of 40 cm and height 1 m. If the outer
side of each of the cones is to be painted and the cost of painting is Rs.25 per m^{2}, what will be the cost of painting
all these cones? (Use π = 3.14 and 1.02.)

Find the volume, curved surface area and the total surface area of a cone having base radius 35 cm and height 12 cm.

_{Find the volume, curved surface area and the total surface area of a cone whose height and slant heights are 6 cm and 10 cm respectively. (Take }_{=3.14)}

A conical pit of diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

**HINT** 1 m^{3} = 1 kilolitre.

A heap of wheat is in the form of a cone of diameter 9 m and height 3.5 m. Find its volume. How much canvas cloth is required to just cover the heap? (Use π = 3.14.)

Radius of a conical heap, r = 4.5 m

Height of a conical tent, h = 3.5 m

A man uses a piece of canvas
having an area of 551 m^{2}, to make a conical tent of base radius 7
m. Assuming that all the stitching margins and wastage incurred while
cutting, amount to approximately 1 m^{2}, find the volume of the tent
that can be made with it.

Radius of a conical tent, r = 7 m

Area
of canvas used in making conical tent = (551 - 1) m^{2} = 550 m^{2}

⇒ Curved surface
area of a conical tent = 550 m^{2}

_{How many meters of cloth , 2.5 m wide , will be required to make conical tent whose base radius is 7 m and height 24 metres?}

_{}

_{Two cones have their height in the ratio 1:3 and the radii of their bases in the ratio3: 1. Show that their volumes are in the ratio 3:1.}

_{A cylinder and a cone have equal radii of their bases and equal height s. If their curved surface areas are in the ratio 8:5, show that the radius and height of each has the ratio 3:4.}

_{A right circular cone is 3.6 cm height and the radius of its base is 1.6 cm. It is melted and recast into a right circular cone having base radius 1.2 cm. Find its height.}

_{A circus tent is cylindrical to a height of 3 meters and conical above it. If its diameter is 105 m and the slant height of the conical portion is 53 m, calculate the length of the canvas 5 m wide to make the required tent.}

_{An iron pillarconsistsof a cylindricalportion2.8 m highand 20cm indiameterand a cone42 cm high is surmounting it . Find the weight of the pillar, given that 1 cm3 of iron weights 7.5 g.}

_{ }

_{From a solid right circular cylinder with height 10 cm and radius of the base 6 cm, a right circular cone of the same height and the base is removed .find the volume of the remaining solid. (Take }_{=3.14)}

_{}

_{Water flows at the rate of 10 meters per minute through a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the surface 40 cm and depth 24 cm?}

A cloth having an area of 165 m^{2}
is shaped into the form of a conical tent of radius 5 m. (i) How many
students can sit in the tent if a student, on an average, occupies m^{2} on the ground? (ii) Find the volume of the
cone.

Curved
surface area of the tent = Area of the cloth = 165 m^{2}

## Chapter 15 - Volume and Surface Area of Solids Exercise Ex. 15A

_{Find the volume, the lateral surface area and the total surface area cuboid whose dimensions are:}

Length =24 m, breadth =25 cmand height =6m

Length = 24 m, breadth = 25 cm =0.25 m, height = 6m.

_{Volume of cuboid= l x b x h}

_{= (24 x 0.25 x 6) m3.}

_{= 36 m3.}

_{Lateral surface area= 2(l + b) x h}

_{= [2(24 +0.25) x 6] m2}

_{= (2 x 24.25 x 6) m2}

_{= 291 m2.}

_{Total surface area =2(lb+ bh + lh)}

_{=2(24 x 0.25+0.25x 6 +24 x 6) m2}

_{= 2(6+1.5+144) m2}

_{= (2 x151.5) m2=303 m2.}

_{Find the volume, the lateral surface area and the total surface area cuboid whose dimensions are:}

Length =15m, breadth =6 m and height =5 dm

Length = 15 m, breadth = 6m and height = 5 dm = 0.5 m

_{Volume of a cuboid = l x b x h}

_{= (15 x 6 x 0.5) m3=45 m3.}

_{Lateral surface area = 2(l + b) x h }

_{= [2(15 + 6) x 0.5] m2}

_{= (2 x 21x0.5) m2=21 m2}

_{Total surface area =2(lb+ bh + lh)}

_{= 2(15 x 6 +6 x 0.5+ 15 x 0.5) m2}

_{= 2(90+3+7.5) m2}

_{= (2 x 100.5) m2}

_{=201 m2}

_{Find the volume, the lateral surface area and the total surface area cuboid whose dimensions are:}

Length =26 m, breadth =14m and height =6.5m

Length 26 m, breadth =14 m and height =6.5 m

_{}Volume of a cuboid= l x b x h

= (26 x 14 x 6.5) m^{3}

= 2366 m^{3}

_{}Lateral surface area of a cuboid =2 (l + b) x h

= [2(26+14) x 6.5] m^{2}

= (2 x 40 x 6.5) m^{2}

= 520 m^{2}

_{Total surface area= 2(lb+ bh + lh)}

_{= 2(26 x 14+14 x6.5 +26 x6.5)}

_{= 2 (364+91+169) m2 }

_{= (2 x 624) m2= 1248 m2.}

_{Find the volume, the lateral surface area and the total surface area cuboid whose dimensions are:}

_{Length=12 cm,breadth=8 cm and height =4.5 cm}

length =12cm, breadth = 8 cm and height = 4.5 cm

_{}Volume of cuboid = l x b x h

= (12 x 8 x 4.5) cm^{3}= 432 cm^{3}

_{}Lateral surface area of a cuboid = 2(l + b) x h

= [2(12 + 8) x 4.5] cm^{2}

= (2 x 20 x 4.5) cm^{2} = 180 cm^{2}

_{}Total surface area cuboid = 2(lb +b h+ l h)

= 2(12 x 8 + 8 x 4.5 + 12 x 4.5) cm^{2}

= 2(96 +36 +54) cm^{2}

= (2 x186) cm^{2}

= 372 cm^{2}

A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What is the volume of a packet containing 12 such matchboxes?

For a matchbox,

Length = 4 cm

Breadth = 2.5 cm

Height = 1.5 cm

Volume of one matchbox = Volume of cuboid

= Length × Breadth × Height

=
(4 × 2.5 × 1.5) cm^{3}

=
15 cm^{3}

Hence,
volume of 12 such matchboxes = 12 × 15 = 180 cm^{3}

A cuboidal water tank is 6 m long,
5 m wide and 4.5 m deep. How many litres of water
can it hold? (Given, 1 m^{3} = 1000 litres.)

For a cuboidal water tank,

Length = 6 m

Breadth = 5 m

Height = 4.5 m

Now,

Volume of a cuboidal water tank = Length × Breadth × Height

=
(6 × 5 × 4.5) m^{3}

=
135 m^{3}

= 135 × 1000 litres

= 135000 litres

Thus, a tank can hold 135000 litres of water.

The capacity of a cuboidal tank is
50000 litres of water. Find the breadth of the tank
if its length and depth are respectively 10 m and 2.5 m. (Given, 1000 litres = 1 m^{3}.)

For a cuboidal water tank,

Length = 10 m

Breadth = 2.5 m

Volume
= 50000 litres = 50 m^{3}

Now,

Volume of a cuboidal tank = Length × Breadth × Height

⇒ 50 = 10 × 2.5 × Height

⇒ Height = 2 m = Depth

Thus, the depth of a tank is 2 m.

A godown measures 40 m × 25 m × 15 m. Find the maximum number of wooden crates, each measuring 1.5 m × 1.25 m × 0.5 m, that can be stored in godown.

For a godown,

Length = 40 m

Breadth = 25 m

Height = 15 m

Volume of a godown = Length × Breadth × Height

= (40 × 25 × 15) m^{3}

For each wooden crate,

Length = 1.5 m

Breadth = 1.25 m

Height = 0.5 m

Volume of each wooden crate = Length × Breadth × Height

= (1.5 × 1.25 × 0.5) m^{3}

_{How many planks of dimensions (5mx25cmX10cm) can be stored in a pit which is 20 m long , 6 m wide and 80 cm deep ?}

_{How many bricks will be required to construct a wall 8 m long , 6 m high and 22.5 cm thick if each brick measures (25cm x11.25cm x 6cm)?}

_{Find the capacity of a closed rectangular cistern whose length is 8 m, breadth 6 m and depth 2.5 m. Also, find the area of the iron sheet required to make the cistern.}

_{Length of Cistern = 8 m}

_{ Breadth of Cistern = 6 m }

_{ And Height (depth) of Cistern =2.5 m }

_{ Capacity of the Cistern = Volume of cistern}

_{ Volume of Cistern = (l x b x h) }

_{ = (8 x 6 x2.5) m3}

_{ =120 m3}

_{ }

_{ }

_{Area of the iron sheet required = Total surface area of the cistem.}

_{ Total surface area = 2(lb +bh +lh)}

_{ = 2(8 x 6 + 6x2.5+ 2.5x8) m2 }

_{ = 2(48 + 15 + 20) m2}

_{ = (2 x 83) m2=166 m2}

The dimensions of a room are (9 m × 8 m × 6.5 m). It has one door of dimensions (2 m × 1.5 m) and two windows, each of dimensions (1.5 m × 1 m). Find the cost of whitewashing the walls at Rs.25 per square metre.

Area of four walls of the room = 2(length + breadth) × Height

=
[2(9 + 8) × 6.5] m^{2}

=
(34 × 6.5) m^{2}

=
221 m^{2}

Area
of one door = Length × Breadth = (2 × 1.5) m^{2} = 3 m^{2}

Area of two windows = 2 × (Length × Breadth)

=
[2 × (1.5 × 1)] m^{2}

=
(2 × 1.5) m^{2}

=
3 m^{2}

Area to be whitewashed

= Area of four walls of the room - Area of one door - Area of two windows

=
(221 - 3 - 3) m^{2}

=
215 m^{2}

Cost of whitewashing = Rs. 25 per square metre

⇒ Cost of
whitewashing 215 m^{2} = Rs. (25 × 215) = Rs. 5375

_{A wall 15 m long , 30 cm wide and 4 m high is made of bricks, each measuring (22cm x12.5cm x7.5cm) if of the total volume of the wall consists of mortar , how many bricks are there in the wall ?}

_{L}

How many cubic centimetres
of iron are there in an open box whose external dimensions are 36 cm, 25 cm,
16.5 cm, the iron being 1.5 cm thick throughout? If
1 cm^{3} of iron weighs 15 g, find the weight of the empty box in
kilograms.

External length of the box = 36 cm

External breadth of the box = 25 cm

External height of the box = 16.5 cm

∴ External
volume of the box = (36 × 25 × 16.5) cm^{3}
= 14850 cm^{3}

Internal length of the box = [36 - (1.5 × 2)] cm = 33 cm

Internal breadth of the box = [25 - (1.5 × 2)] cm = 22 cm

Internal height of the box = (16.5 - 1.5) cm = 15 cm

∴ Internal
volume of the box = (33 × 22 × 15) cm^{3}
= 10890 cm^{3}

Thus, volume of iron used in the box

= External volume of the box - Internal volume of the box

=
(14850 - 10890) cm^{3}

=
3960 cm^{3}

A box made of sheet metal costs Rs.6480 at Rs.120 per square metre. If the box is 5 m long and 3 m wide, find its height.

_{The volume of a cuboid is 1536m3. Its length is 16m, and its breadth and height are in the ratio 3:2. Find the breadth and height of the cuboid.}

_{How many persons can be accommodated in a dining hall of dimensions (20m x16mx4.5m), assuring that each person's requires 5 cubic metres of air?}

_{A classroom is 10m long, 6.4 m wide and 5m high. If each student be given 1.6 m2 of the floor area, how many students can be accommodated in the room? How many cubic metres of air would each student get?}

_{The surface of the area of a cuboid is 758 cm2. Its length and breadth are 14 cm and 11cm respectively. Find its height.}

Find the volume, the lateral surface area, the total surface area and the diagonal of cube, each of whose edges measures 9m. [Take ]

_{The total surface area of a cube is 1176 cm2. Find its volume.}

_{The lateral surface area of a cube is 900 cm2. Find its volume.}

_{ }

_{The volume of a cube is 512 cm3. Find its surface area.}

_{Three cubes of metal with edges 3cm, 4 cm and 5 cm respectively are melted to form a single cube. Find the lateral surface area of the new cube formed.}

_{Find the length of the longest pole that can be put in a room of dimensions (10mx 10m x5m).}

The sum of length, breadth and depth of a cuboid is 19 cm and length of its diagonal is 11 cm. Find the surface area of the cuboid.

Each edge of a cube is increased by 50%. Find the percentage increase in the surface area of the cube.

Let the edge of the cube = 'a' cm

Then,
surface area of cube = 6a^{2} cm^{2}

If V is the volume of a cuboid of dimensions a, b, c and S is its surface area then prove that

Water in a canal, 30 dm wide and 12 dm deep, is flowing with a velocity of 20 km per hour. How much area will it irrigate, if 9 cm of standing water is desired?

A solid metallic cuboid of dimensions (9 m × 8 m × 2 m) is melted and recast into solid cubes of edge 2 m. Find the number of cubes so formed.

Volume
of a cuboid = (9 × 8 × 2) m^{3}
= 144 m^{3}

Volume
of each cube of edge 2 m = (2 m)^{3} = 8 m^{3}

## Chapter 15 - Volume and Surface Area of Solids Exercise Ex. 15B

_{The diameter of a cylinder is 28 cm and its height is 40 cm. find the curved surface area, total surface area and the volume of the cylinder.}

_{ }

_{ }

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients?

Radius (r) of cylindrical bowl =

Height (h) up to which the bowl is filled with soup = 4 cm

Volume of soup in 1 bowl
= pr^{2}h = 154 cm^{3}

Hence, volume of soup in
250 bowls = (250 × 154) cm^{3 }= 38500 cm^{3
}= 38.5 litres

Thus, the hospital will have to prepare 38.5 litres of soup daily to serve 250 patients.

The pillars of a temple are cylindrically shaped. Each pillar has a circular base of radius 20 cm and height 10 m. How much concrete mixture would be required to build 14 such pillars?

Radius (r) of pillar = 20 cm = m

Height (h) of pillar = 10 m

A soft drink is available in two packs: (i) a tin can with a rectangular base of length 5 cm, breadth 4 cm and height 15 cm, and (ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?

For a tin can of rectangular base,

Length = 5 cm

Breadth = 4 cm

Height = 15 cm

∴ Volume of a tin can = Length × Breadth × Height

=
(5 × 4 × 15) cm^{3}

=
300 cm^{3}

For a cylinder with circular base,

Diameter = 7 ⇒ Radius = r = cm

Height = h = 10 cm

⇒ Volume of plastic cylinder is greater than volume of a tin can.

Difference
in volume = (385 - 300) = 85 cm^{3}

^{ }

Thus,
a plastic cylinder has more capacity that a tin can by 85 cm^{3}.

There are 20 cylindrical pillars
in a building, each having a diameter of 50 cm and height 4 m. Find the cost
of cleaning them at Rs.14 per m^{2}.

Radius (r) of 1 pillar =

Height (h) of 1 pillar = 4 m

The curved surface area of a right
circular cylinder is 4.4 m^{2}. If the radius of its base is 0.7 m,
find its (i) height and (ii) volume.

Curved surface area of a
cylinder = 4.4 m^{2}

Radius (r) of a cylinder = 0.7 m

The lateral surface area of a
cylinder is 94.2 cm^{2} and its height is 5 cm. Find (i) the radius of its base and (ii) its volume. (Take π =
3.14.)

Lateral surface area of
a cylinder = 94.2 cm^{2}

Height (h) of a cylinder = 5 cm

The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. Find the area of the metal sheet needed to make it.

Volume of a cylinder =
15.4 litres = 15400 cm^{3}

Height (h) of a cylinder = 1 m = 100 cm

The inner diameter of a
cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length
of the pipe is 35 cm. Find the mass of the pipe, if 1 cm^{3} of wood
has a mass of 0.6 g.

Internal diameter of a cylinder = 24 cm

⇒ Internal radius of a cylinder, r = 12 cm

External diameter of a cylinder = 28 cm

⇒ External radius of a cylinder, R = 14 cm

Length of the pipe, i.e height, h = 35 cm

In a water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.

Diameter of a cylindrical pipe = 5 cm

⇒ Radius (r) of a cylindrical pipe = 2.5 cm

Height (h) of a cylindrical pipe = 28 m = 2800 cm

_{Find the weight of a solid cylinder of radius10.5 cm and height 60 cm if the material of the cylinder weights 5 g per cm2}

_{The curved surface area of a cylinder is 1210 cm2 and its diameter is 20 cm. find its height and volume.}

_{ }

_{The curved surface area of a cylinder is 4400 cm}^{2}_{ and the circumferences of its base are 110 cm. Find the height and the volume of the cylinder.}

_{The radius of the base and the height of a cylinder are in the ratio 2:3. If its volume is 1617 cm3, find the total surface area of the cylinder}

The total surface area of the cylinder is 462 cm^{2}. And its curved surface area is one third of its total surface area. Find the volume of the cylinder.

_{The total surface area of the solid cylinder is 231 cm2 and its curved surface area is of the total surface area. Find the volume of the cylinder.}

_{}

_{The ratio between the curved surface area and the total surface area of a right circular cylinder is 1:2. Find the volume of the cylinder if its total surface area is 616 cm2.}

_{}

_{}

_{A cylindrical bucket , 28 cm in diameter and 72 cm high , is full of water .The water is emptied into a rectangular tank, 66 cm long and 28 cm wide. Find the height of the water level in the tank}

_{The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up on writing 330 words on an average. How many words would use up a bottle of ink containing one fifth of a liter?}

_{1 cm3 of gold is drawn into a wire 0.1 mm is diameter. Find the length of a wire.}

_{Ifs 1 cm3 of cast iron weighs 21 g, find the weight of a cast iron pipe of length 1 m with a bore of 3 cm in which the thickness of the metal is 1 cm.}

_{}

_{A cylindrical tube, open at both ends, is made of metal. The internal diameter of the tube is 10.4 cm and its length is 25 cm. The thickness of the metal is 8 mm everywhere. Calculate the volume of the metal.}

It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same?

Diameter of a cylinder = 140 cm

⇒ Radius, r = 70 cm

Height (h) of a cylinder = 1 m = 100 cm

A juiceseller has a large cylindrical vessel of base radius 15 cm filled up to a height of 32 cm with orange juice. The juice is filled in small cylindrical glasses of radius 3 cm up to a height of 8 cm, and sold for Rs.15 each, How much money does he receive by selling the juice completely?

Radius (r) of cylindrical vessel = 15 cm

Height (h) of cylindrical vessel = 32 m

Radius of small cylindrical glass = 3 cm

Height of a small cylindrical glass = 8 cm

A well with inside diameter 10 m is dug 8.4 m deep. Earth taken out of it is spread all around it to a width of 7.5 m to form an embankment. Find the height of the embankment.

Radius of the well = 5 m

Depth of the well = 8.4 m

Width of the embankment = 7.5 m

External radius of the embankment, R = (5 + 7.5) m = 12.5 m

Internal radius of the embankment, r = 5 m

Area
of the embankment = π (R^{2} - r^{2})

Volume
of the embankment = Volume of the earth dug out = 660 m^{2}

How many litres
of water flows out of a pipe having an area of cross section of 5 cm^{2}
in 1 minute, if the speed of water in the pipe is 30 cm/sec?

Speed of water = 30 cm/sec

∴ Volume of water that flows out of the pipe in one second

= Area of cross-section × Length of water flown in one second

=
(5 × 30) cm^{3}

=
150 cm^{3}

Hence, volume of water that flows out of the pipe in 1 minute

=
(150 × 60) cm^{3}

=
9000 cm^{3}

= 9 litres

A cylindrical water tank of diameter 1.4 m and height 2.1 m is being fed by a pipe of diameter 3.5 cm through which water flows at the rate of 2 m per second. In how much time will the tank be filled?

Suppose the tank is filled in x minutes. Then,

Volume of the water that flows out through the pipe in x minutes

= Volume of the tank

Hence, the tank will be filled in 28 minutes.

A cylindrical container with diameter of base 56 cm contains sufficient water to submerge a rectangular solid of iron with dimensions (32 cm × 22 cm × 14 cm). Find the rise in the level of water when the solid is completely submerged.

Let the rise in the level of water = h cm

Then,

Volume of the cylinder of height h and base radius 28 cm

= Volume of rectangular iron solid

Thus, the rise in the level of water is 4 cm.

Find the cost of sinking a tube-well 280 m deep, having a diameter 3 m at the rate of Rs.15 per cubic metre. Find also the cost of cementing its inner curved surface at Rs.10 per square metre.

Radius, r = 1.5 m

Height, h = 280 m

Find the length of 13.2 kg of copper wire of diameter 4 mm, when 1 cubic centimetre of copper weights 8.4 g.

Let the length of the wire = 'h' metres

Then,

Volume of the wire × 8.4 g = (13.2 × 1000) g

Thus, the length of the wire is 125 m.

It costs Rs.3300 to paint the inner curved surface of a cylindrical
vessel 10 m deep at the rate of Rs.30 per m^{2}. Find the

(i) inner curved surface area of the vessel,

(ii) inner radius of the base, and

(iii) capacity of the vessel.

The difference between inside and
outside surfaces of a cylindrical tube 14 cm long, is 88 cm^{2}. If
the volume of the tube is 176 cm^{3}, find the inner and outer radii
of the tube.

Let R cm and r cm be the outer and inner radii of the cylindrical tube.

We have, length of tube = h = 14 cm

Now,

Outside
surface area - Inner surface area = 88 cm^{2}

⇒ 2πRh - 2πrh = 88

⇒ 2π(R - r)h = 88

It
is given that the volume of the tube = 176 cm^{3}

⇒ External
volume - Internal volume = 176 cm^{3}

⇒ πR^{2}h
- πr^{2}h = 176

⇒ π (R^{2}
- r^{2})h = 176

Adding (i) and (ii), we get

2R = 5

⇒ R = 2.5 cm

⇒ 2.5 - r = 1

⇒ r = 1.5 cm

Thus, the inner and outer radii of the tube are 1.5 cm and 2.5 cm respectively.

A rectangular sheet of paper 30 cm × 18 cm can be transformed into the curved surface of a right circular cylinder in two ways namely, either by rolling the paper along its length or by rolling it along its breadth. Find the ratio of the volumes of the two cylinders, thus formed.

When
the sheet is folded along its length, it forms a cylinder of height, h_{1}
= 18 cm and perimeter of base equal to 30 cm.

Let
r_{1} be the radius of the base and V_{1} be is volume.

Then,

Again,
when the sheet is folded along its breadth, it forms a cylinder of height, h_{2}
= 30 cm and perimeter of base equal to 18 cm.

Let
r_{2} be the radius of the base and V_{2} be is volume.

Then,

## Chapter 15 - Volume and Surface Area of Solids Exercise Ex. 15D

Find the volume and the surface area of a sphere whose radius is:

5 m

Find the volume and the surface area of a sphere whose radius is

4.2 cm

Find the volume and the surface area of a sphere whose radius is

3.5 cm

_{The volume of a sphere is 38808 cm3. Find the radius and hence its surface area.}

_{Find the surface area of a sphere whose volume is 606.375 m3}

Find the volume of a sphere whose
surface area is 154 cm^{2}.

Surface
area of sphere = 154 cm^{2}

⇒ 4πr^{2}
= 154

_{The surface area of a sphere is (576) cm2. Find its volume.}

_{How many leads shots, each 3 mm in diameter, can be made from cuboid with dimensions (12cm x 11cm x 9cm)?}

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

_{A solid sphere of radius 3 cm is melted and then cast into smaller spherical balls, each of diameters 0.6 cm. find the number of small balls thus obtained.}

_{A metallic sphere of radius 10.5 cm is melted an then recast into smaller cones , each of radius 3.5 cm and height 3 cm. How many cones are obtained?}

_{How many spheres 12 cm in diameter can be made from a metallic cylinder of diameter 8 cm and }

_{height 90 cm ?}

_{The diameter of sphere is 6 cm. It is melted and drawn into wire of diameter 2 mm. Find the length of the wire.}

_{The diameter of the copper sphere is 18cm. 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.}

_{A sphere of a diameter 15.6 cm is melted and cast into a right circular cone of height 31.2 cm. find the diameter of the base of the cone.}

_{A spherical cannonball 28 cm in diameter is melted and recast into a right circular cone mould, whose base is 35 cm in diameter. Find the height of the cone.}

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

_{}

The radii of two spheres are in the ratio 1:2. Find the ratio of their surface areas.

The surface areas of two spheres are in the ratio 1:4. Find the ratio of their volumes.

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

A cylindrical bucket with base radius 15 cm is filled with water to up height of 20 cm. a heavy iron spherical ball of radius 9 cm is dropped into the bucket to submerge completely in the water . Find the increase in the level of water

_{The outer diameter of a spherical shell is 12 cm and its inner diameter is 8 cm. Find the volume of metal contained in the shell. Also, find its outer surface area.}

_{}

A hollow spherical shell is made of a metal of density 4.5 g per cm^{3}. If it's internal and external radii are 8 cm and 9cm respectively, find the weight of the shell.

_{}

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

_{}

A hemisphere bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into cylindrical shaped bottles of diameter 3 cm and height 4 cm. How many bottles are required to empty the bowl?

A hemispherical bowl is made of steel 0.5 cm thick. The inside radius of the bowl is 4 cm. Find the volume of steel used in making the bowl.

A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.

Inner radius = 5 cm

⇒ Outer radius = 5 + 0.25 = 5.25 cm

A hemispherical bowl made of brass
has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at
the rate of Rs.32
per 100 cm^{2}.

Inner diameter of the hemispherical bowl = 10.5 cm

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?

Let the diameter of earth = d

⇒ Radius of the earth =

Then, diameter of moon = .

⇒ Radius of moon =

Volume of moon

Volume of earth

Thus, the volume of moon is of volume of earth.

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of the hemisphere?

Volume of a solid hemisphere = Surface area of a solid hemisphere

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