Chapter 15 : Volume and Surface Area of Solids - R S Aggarwal And V Aggarwal Solutions for Class 9 Maths CBSE

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Chapter 15 - Volume and Surface Area of Solids Excercise MCQ

Question 1

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

  1. 243 cm3
  2. 405 cm3
  3. 810 cm3
  4. 603 cm3
Solution 1

Question 2

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

  1. 864 cm2
  2. 552 cm2
  3. 432 cm2
  4. 276 cm2
Solution 2

Question 3

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

  1. 45 m2
  2. 21 m2
  3. 201 m2
  4. 90 m2
Solution 3

Question 4

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

  1. 27 kg
  2. 48 kg
  3. 36 kg
  4. 56 kg
Solution 4

Question 5

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

  1. 15 m
  2. 16 m
  3.  
  4. 12 m
Solution 5

Question 6

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

  1. 8 cm
  2. 9.5 cm
  3. 19 cm
  4. 11.2 cm
Solution 6

Question 7

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

  1. 190
  2. 192
  3. 184
  4. 180
Solution 7

Question 8

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?

  1. 480
  2. 450
  3. 320
  4. 360
Solution 8

Question 9

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)?

  1. 4800
  2. 5600
  3. 6400
  4. 5200
Solution 9

Question 10

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 m3 of air?

  1. 250
  2. 270
  3. 320
  4. 300
Solution 10

Question 11

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

  1. 2000 m3
  2. 2250 m3
  3. 2500 m3
  4. 2750 m3
Solution 11

Question 12

The lateral surface area of a cube is 256 m2. The volume of the cube is

  1. 64 m3
  2. 216 m3
  3. 256 m3
  4. 512 m3
Solution 12

Question 13

The total surface area of a cube is 96m2. The volume of the cube is

  1. 8 cm3
  2. 27cm3
  3. 64cm3
  4. 512 cm3
Solution 13

Question 14

The volume of a cube is 512 cm3. Its surface area is

  1. 256 cm2
  2. 384 cm2
  3. 512 cm2
  4. 64 cm2
Solution 14

Question 15

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

  1. 10 cm
  2. 20 cm
  3.  
Solution 15

Question 16

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

  1. 192 cm2
  2. 384 cm2
  3. 512 cm2
  4. 768 cm2
Solution 16

Question 17

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

  1. 50%
  2. 75%
  3. 100%
  4. 125%
Solution 17

Question 18

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

  1. 72 cm2
  2. 144 cm2
  3. 128 cm2
  4. 256 cm2
Solution 18

Question 19

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

  1. 500 m3
  2. 750 m3
  3. 800 m3
  4. 1000 m3
Solution 19

Question 20

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

  1. 1:3
  2. 1:8
  3. 1:9
  4. 1:18
Solution 20

  

Question 21

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

  1. is doubled
  2. becomes 4 times
  3. becomes 6 times
  4. becomes 8 times
Solution 21

  

Question 22

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 cm3

b. 396 cm3

c. 495 cm3

d. 297 cm3

Solution 22

  

Question 23

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

  1. 880 cm2
  2. 1760 cm2
  3. 3520 cm2
  4. 2640 cm2
Solution 23

  

Question 24

If the curved surface area of a cylinder is 1760 cm2 and its base radius is 14 cm, then its height is

  1. 10 cm
  2. 15 cm
  3. 20 cm
  4. 40 cm
Solution 24

  

Question 25

The height of a cylinder is 14 cm and its curved surface area is 264 cm2. The volume of the cylinder is

  1. 308 cm2
  2. 396 cm2
  3. 1232 cm2
  4. 1848 cm2
Solution 25

  

Question 26

The curved surface area of the cylindrical pillar is 264 m2 and its volume is 924m3. The height of the pillar is

  1. 4 m
  2. 5 m
  3. 6 m
  4. 7 m
Solution 26

  

Question 27

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

  1. 2:5
  2. 8:7
  3. 10:9
  4. 16:9
Solution 27

  

Question 28

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

  1. 27:20
  2. 20:27
  3. 4:9
  4. 9:4
Solution 28

  

Question 29

The ratio between the radius of the base and height of a cylinder is 2:3. If its volume is 1617 cm3, then its total surface area is

  1. 308 cm2
  2. 462 cm2
  3. 540 cm2
  4. 770 cm2
Solution 29

  

Question 30

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

Solution 30

  

Question 31

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 cm2, then the volume of the cylinder is

  1. 1078 cm3
  2. 1232 cm3
  3. 1848 cm3
  4. 924 cm3
Solution 31

  

Question 32

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

  1. The same
  2. Doubled
  3. Halved
  4. Four times
Solution 32

  

Question 33

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

  1. 540
  2. 450
  3. 380
  4. 472
Solution 33

  

Question 34

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

  1. 2 times
  2. 3 times
  3. 6 times
  4. 9 times
Solution 34

  

Question 35

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

  1. 1440 m2
  2. 1320 m2
  3. 1260 m2
  4. 1550 m2
Solution 35

  

Question 36

2.2 dm3 of lead is to be drawn into a cylindrical wire 0.50 cm in diameter. The length of the wire is

  1. 110 m
  2. 112 m
  3. 98 m
  4. 124 m
Solution 36

  

Question 37

The lateral surface area of a cylindrical is

Solution 37

  

Question 38

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

  1. 528 cm2
  2. 550 cm2
  3. 616 cm2
  4. 704 cm2
Solution 38

  

Question 39

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

  1. (12π) cm3
  2. (36π) cm3
  3. (72π) cm3
  4. (144π) cm3
Solution 39

  

Question 40

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

  1. 120 m
  2. 180 m
  3. 220 m
  4. 550 m
Solution 40

  

Question 41

The volume of a cone is 1570 cm3 and its height is 15 cm. What is the radius of the cone? (Use π = 3.14)

  1. 10 cm
  2. 9 cm
  3. 12 cm
  4. 8.5 cm
Solution 41

  

Question 42

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

  1. 7356 cm3
  2. 7546 cm3
  3. 7506 cm3
  4. 7564 cm3
Solution 42

Correct option: (b)

Question 43

The volume of a right circular cone of height 24 cm is 1232 cm3. Its curved surface area is

  1. 1254 cm2
  2. 704 cm2
  3. 550 cm2
  4. 462 cm2
Solution 43

  

Question 44

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. 1:5
  2. 5:4
  3. 25:16
  4. 25:64
Solution 44

  

Question 45

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

  1. 100%
  2. 200 %
  3. 300 %
  4. 400 %
Solution 45

  

Question 46

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

  1. 2:1
  2. 4:1
  3. 8:1
  4. 1:1
Solution 46

  

Question 47

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. 1:3
  2. 3:1
  3. 4:3
  4. 3:4
Solution 47

  

Question 48

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

  1. 3:5
  2. 2:5
  3. 3:1
  4. 1:3
Solution 48

  

Question 49

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

  1. 9:8
  2. 8:9
  3. 3:4
  4. 4:3
Solution 49

  

Question 50

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

  1. 3 times
  2. 4 times
  3. 6 times
  4. 8 times
Solution 50

  

Question 51

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

  1. 450
  2. 1350
  3. 4500
  4. 13500
Solution 51

  

Question 52

A conical tent is to accommodate 11 persons such that each person occupies 4 m2 of space on the ground. They have 220m3 of air to breathe. The height of the cone is

  1. 14m
  2. 15 m
  3. 16 m
  4. 20 m
Solution 52

  

Question 53

The volume of a sphere of radius 2r is

Solution 53

  

Question 54

The volume of a sphere of a radius 10.5 cm is

  1. 9702 cm3
  2. 4851 cm3
  3. 19404 cm3
  4. 14553 cm3
Solution 54

  

Question 55

The surface area of a sphere of radius 21 cm is

  1. 2772 cm2
  2. 1386 cm2
  3. 4158 cm2
  4. 5544 cm2
Solution 55

  

Question 56

The surface area of a sphere is 1386 cm2. Its volume is

  1. 1617 cm3
  2. 3234 cm3
  3. 4851 cm3
  4. 9702 cm3
Solution 56

  

Question 57

If the surface area of a sphere is (144 π) m2, then its volume is

  1. (288 π) m3
  2. (188 π) m3
  3. (300 π) m3
  4. (316 π) m3
Solution 57

  

Question 58

The volume of a sphere is 38808 cm3. Its curved surface area is

  1. 5544 cm2
  2. 8316 cm2
  3. 4158 cm2
  4. 1386 cm2
Solution 58

  

Question 59

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

  1. 1:2
  2. 1:4
  3. 1:8
  4. 1:16
Solution 59

  

Question 60

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

  1. 8
  2. 16
  3. 32
  4. 64
Solution 60

  

Question 61

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

  1. 4.2 cm
  2. 2.1 cm
  3. 2.4 cm
  4. 1.6 cm
Solution 61

  

Question 62

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

  1. 272 m
  2. 288 m
  3. 292 m
  4. 296 m
Solution 62

  

Question 63

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

  1. 21
  2. 63
  3. 126
  4. 130
Solution 63

  

Question 64

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

  1. 7200
  2. 8400
  3. 72000
  4. 84000
Solution 64

  

Question 65

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

  1. 12 m
  2. 18 m
  3. 36 m
  4. 66 m
Solution 65

  

Question 66

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

  1. 6.3 cm
  2. 2.1 cm
  3. 6 cm
  4. 4 cm
Solution 66

  

Question 67

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. 1 cm
  2. 1.5 cm
  3. 2.5 cm
  4. 0.5 cm
Solution 67

  

Question 68

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. 1:4
  2. 1:3
  3. 2:3
  4. 1:2
Solution 68

Question 69

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

  1. 38 cm2
  2. 58 cm2
  3. 78 cm2
  4. 88 cm2
Solution 69

Question 70

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?

  1. 27
  2. 35
  3. 54
  4. 63
Solution 70

Question 71

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

  1. 1:2
  2. 2:1
  3. 4:1
Solution 71

Question 72

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

  1. 1:2:3
  2. 2:1:3
  3. 2:3:1
  4. 3:2:1
Solution 72

Question 73

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

  1. 1 units
  2. 2 units
  3. 3 units
  4. 4 units
Solution 73

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

Question 1

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

Solution 1

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

Volume of cuboid = l x b x h

= (12 x 8 x 4.5) cm3= 432 cm3

 

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

= [2(12 + 8) x 4.5] cm2

= (2 x 20 x 4.5) cm2 = 180 cm2

 

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

= 2(12 x 8 + 8 x 4.5 + 12 x 4.5) cm2

= 2(96 +36 +54) cm2

= (2 x186) cm2

= 372 cm2

 

Question 2

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

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

Solution 2

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.

Question 3

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

Solution 3

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

Question 4

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

Solution 4

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) m3

= 2366 m3

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

= [2(26+14) x 6.5] m2

= (2 x 40 x 6.5) m2

= 520 m2

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.

Question 5

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

Solution 5

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) cm3

= 15 cm3

 

Hence, volume of 12 such matchboxes = 12 × 15 = 180 cm3

Question 6

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 m3 = 1000 litres.)

Solution 6

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) m3

= 135 m3

= 135 × 1000 litres

= 135000 litres

 

Thus, a tank can hold 135000 litres of water. 

Question 7

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 m3.)

Solution 7

For a cuboidal water tank,

Length = 10 m

Breadth = 2.5 m

Volume = 50000 litres = 50 m3

 

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. 

Question 8

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.

Solution 8

For a godown,

Length = 40 m

Breadth = 25 m

Height = 15 m

Volume of a godown = Length × Breadth × Height

= (40 × 25 × 15) m3

 

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) m3

 

Question 9

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

Solution 9

Question 10

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)?

Solution 10

Question 11

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.

Solution 11

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

Question 12

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.

Solution 12

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

= [2(9 + 8) × 6.5] m2

= (34 × 6.5) m2

= 221 m2

 

Area of one door = Length × Breadth = (2 × 1.5) m2 = 3 m2

 

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

= [2 × (1.5 × 1)] m2

= (2 × 1.5) m2

= 3 m2

 

Area to be whitewashed

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

= (221 - 3 - 3) m2

= 215 m2

 

Cost of whitewashing = Rs. 25 per square metre

Cost of whitewashing 215 m2 = Rs. (25 × 215) = Rs. 5375

Question 13

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 ?

Solution 13

L

Question 14

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 cm3 of iron weighs 15 g, find the weight of the empty box in kilograms.

Solution 14

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) cm3 = 14850 cm3

 

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) cm3 = 10890 cm3

 

Thus, volume of iron used in the box

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

= (14850 - 10890) cm3

= 3960 cm3

 

Question 15

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.

Solution 15

  

Question 16

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.

Solution 16

Question 17

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?

Solution 17
Question 18

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?

Solution 18


Question 19

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

Solution 19

Question 20
In  shower, 5 cm of rain falls. Find the volume of water that falls on 2 hectares of ground.
Solution 20
Question 21

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

Solution 21

Question 22

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

Solution 22

Question 23

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

Solution 23

 

Question 24

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

Solution 24

Question 25

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.

Solution 25

Question 26

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

Solution 26

Question 27

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.

Solution 27

Question 28

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

Solution 28

Let the edge of the cube = 'a' cm

Then, surface area of cube = 6a2 cm2

 

Question 29

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

Solution 29

  

Question 30

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?

Solution 30

Question 31

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.

Solution 31

Volume of a cuboid = (9 × 8 × 2) m3 = 144 m3

Volume of each cube of edge 2 m = (2 m)3 = 8 m3

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

Question 1

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.

Solution 1

 


Question 2

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?

Solution 2

Radius (r) of cylindrical bowl =

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

Volume of soup in 1 bowl = pr2h   = 154 cm3

Hence, volume of soup in 250 bowls = (250 × 154) cm3 = 38500 cm3 = 38.5 litres

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

Question 3

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?

Solution 3

Radius (r) of pillar = 20 cm =   m

Height (h) of pillar = 10 m

Question 4

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?

Solution 4

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) cm3

= 300 cm3

 

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 cm3

 

Thus, a plastic cylinder has more capacity that a tin can by 85 cm3.

Question 5

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 m2.

Solution 5

Radius (r) of 1 pillar =

Height (h) of 1 pillar = 4 m

  

Question 6

The curved surface area of a right circular cylinder is 4.4 m2. If the radius of its base is 0.7 m, find its (i) height and (ii) volume.

Solution 6

Curved surface area of a cylinder = 4.4 m2

Radius (r) of a cylinder = 0.7 m

 

  

Question 7

The lateral surface area of a cylinder is 94.2 cm2 and its height is 5 cm. Find (i) the radius of its base and (ii) its volume. (Take π = 3.14.)

Solution 7

Lateral surface area of a cylinder = 94.2 cm2

Height (h) of a cylinder = 5 cm

 

  

Question 8

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.

Solution 8

Volume of a cylinder = 15.4 litres = 15400 cm3

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

 

  

Question 9

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 cm3 of wood has a mass of 0.6 g.

Solution 9

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

Question 10

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.

Solution 10

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

 

  

Question 11

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

Solution 11
Question 12

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

Solution 12

Question 13

The curved surface area of a cylinder is 4400 cm2 and the circumferences of its base are 110 cm. Find the height and the volume of the cylinder.

Solution 13
Question 14

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

Solution 14

Question 15

The total surface area of the cylinder is 462 cm2. And its curved surface area is one third of its total surface area. Find the volume of the cylinder.

Solution 15

 

 

Question 16

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.

Solution 16

Question 17

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.

Solution 17

Question 18

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

Solution 18

Question 19

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?

Solution 19

Question 20

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

Solution 20

Question 21

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.

Solution 21

Question 22

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.

Solution 22

Question 23

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?

Solution 23

Diameter of a cylinder = 140 cm

Radius, r = 70 cm

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

 

  

Question 24

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?

Solution 24

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

Question 25

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.

Solution 25

  

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 (R2 - r2)

Volume of the embankment = Volume of the earth dug out = 660 m2

Question 26

How many litres of water flows out of a pipe having an area of cross section of 5 cm2 in 1 minute, if the speed of water in the pipe is 30 cm/sec?

Solution 26

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) cm3

= 150 cm3

 

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

= (150 × 60) cm3

= 9000 cm3

= 9 litres

Question 27

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?

Solution 27

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.

Question 28

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.

Solution 28

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.

Question 29

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.

Solution 29

Radius, r = 1.5 m

Height, h = 280 m

 

Question 30

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

Solution 30

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.

Question 31

It costs Rs.3300 to paint the inner curved surface of a cylindrical vessel 10 m deep at the rate of Rs.30 per m2. Find the

(i) inner curved surface area of the vessel,

(ii) inner radius of the base, and

(iii) capacity of the vessel.

 

Solution 31

Question 32

The difference between inside and outside surfaces of a cylindrical tube 14 cm long, is 88 cm2. If the volume of the tube is 176 cm3, find the inner and outer radii of the tube.

Solution 32

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 cm2

2πRh - 2πrh = 88

2π(R - r)h = 88

 

It is given that the volume of the tube = 176 cm3

External volume - Internal volume = 176 cm3

πR2h - πr2h = 176

π (R2 - r2)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.

Question 33

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.

Solution 33

When the sheet is folded along its length, it forms a cylinder of height, h1 = 18 cm and perimeter of base equal to 30 cm.

Let r1 be the radius of the base and V1 be is volume.

Then,

 

Again, when the sheet is folded along its breadth, it forms a cylinder of height, h2 = 30 cm and perimeter of base equal to 18 cm.

Let r2 be the radius of the base and V2 be is volume.

Then,

 

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

Question 1

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

Solution 1

Radius of a cone, r = 5.25 cm

Slant height of a cone, l = 10 cm

Question 2

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

Solution 2

Radius of a cone, r = 12 m

Slant height of a cone, l = 21 cm

  

Question 3

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.

Solution 3

Radius of a conical cap, r = 7 cm

Height of a conical cap, h = 24 cm

Thus, 5500 cm2 sheet will be required to make 10 caps.

Question 4

The curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find the radius of the base and total surface area of the cone.

Solution 4

Let r be the radius of a cone.

Slant height of a cone, l = 14 cm

Curved surface area of a cone = 308 cm2

 

Question 5

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 m2.

Solution 5

Radius of a cone, r = 7 m

Slant height of a cone, l = 25 m

Cost of whitewashing = Rs. 12 per m2

Cost of whitewashing 550 m2 area = Rs. (12 × 550) = Rs. 6600 

Question 6

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 m2 canvas is Rs.70, find the cost of canvas required to make the tent.

Solution 6

Radius of a conical tent, r = 24 m

Height of a conical tent, h = 10 m

Question 7

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 m2, what will be the cost of painting all these cones? (Use π = 3.14 and  1.02.)

Solution 7

  

Question 8

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

Solution 8

  

Question 9

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)

Solution 9

Question 10

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

HINT 1 m3 = 1 kilolitre.

Solution 10

Question 11

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.)

Solution 11

Radius of a conical heap, r = 4.5 m

Height of a conical tent, h = 3.5 m

  

Question 12

A man uses a piece of canvas having an area of 551 m2, 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 m2, find the volume of the tent that can be made with it.

Solution 12

Radius of a conical tent, r = 7 m

Area of canvas used in making conical tent = (551 - 1) m2 = 550 m2

Curved surface area of a conical tent = 550 m2

  

Question 13

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?

Solution 13

Question 14

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.

Solution 14

Question 15

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.

Solution 15

Question 16

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.

Solution 16

Question 17

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.

Solution 17

Question 18

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.

Solution 18

 

Question 19

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)

Solution 19

Question 20

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?

Solution 20

Question 21

A cloth having an area of 165 m2 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  m2 on the ground? (ii) Find the volume of the cone.

Solution 21

Curved surface area of the tent = Area of the cloth = 165 m2

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

Question 1

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

5 m

Solution 1

 

Question 2

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

4.2 cm

Solution 2

 


Question 3

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

3.5 cm

Solution 3

 

Question 4

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

Solution 4

Question 5

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

Solution 5

Question 6

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

Solution 6

Surface area of sphere = 154 cm2

4πr2 = 154

Question 7

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

Solution 7

Question 8

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

Solution 8

Question 9

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

Solution 9

Question 10

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.

Solution 10

Question 11

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?

Solution 11

Question 12

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

height 90 cm ?

Solution 12

Question 13

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

Solution 13

Question 14

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.

Solution 14

Question 15

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.

Solution 15

Question 16

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.

Solution 16

Question 17

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.

Solution 17

Question 18

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

Solution 18

Question 19

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

Solution 19

Question 20

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?

Solution 20

Question 21

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

Solution 21

Question 22

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.

Solution 22

Question 23

A hollow spherical shell is made of a metal of density 4.5 g per cm3. If it's internal and external radii are 8 cm and 9cm respectively, find the weight of the shell.

Solution 23

Question 24

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.

Solution 24

Question 25

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?

Solution 25

Question 26

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.

Solution 26

We space have comma
straight r equals Inner space radius space of space the space bowl equals 4 space cm
straight R equals outer space radius space of space the space bowl equals left parenthesis 4 plus 0.5 right parenthesis space cm equals 4.5 space cm
Volume space of space the space inner space hemisphere equals 2 over 3 πr cubed equals 2 over 3 cross times 22 over 7 cross times 4 cross times 4 space cm cubed
Volume space of space the space outer space hemisphere equals 2 over 3 πR cubed equals 2 over 3 cross times 22 over 7 cross times 4.5 cross times 4.5 space cm cubed
therefore space Volume space of space steel space used
space space space space space equals Volume space of space the space outer space hemisphere minus Volume space of space the space inner space hemisphere
space space space space space equals 2 over 3 cross times 22 over 7 cross times 4.5 cross times 4.5 minus 2 over 3 cross times 22 over 7 cross times 4 cross times 4
space space space space space equals 2 over 3 cross times 22 over 7 cross times open square brackets open parentheses 4.5 close parentheses squared minus open parentheses 4 close parentheses squared close square brackets
space space space space space equals 44 over 21 cross times open parentheses 91.125 minus 64 close parentheses
space space space space space equals 44 over 21 cross times 27.125
space space space space space equals 56.83 space space cm cubed

Question 27

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.

Solution 27

Inner radius = 5 cm

Outer radius = 5 + 0.25 = 5.25 cm

Question 28

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 cm2.

Solution 28

Inner diameter of the hemispherical bowl = 10.5 cm

 

Question 29

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?

Solution 29

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.

Question 30

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

Solution 30

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

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