RS Agarwal Solution | Class 9th | Chapter-15 |  Volumes and Surface Area of Solids | Edugrown

Exercise 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.  
4. 

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 cm³Solution 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
4. 

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

Exercise 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

Exercise Ex. 15A

Question 1(iv)

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

Length =24 m, breadth =25 cmand height =6mSolution 1(iv)

_{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 1(iii)

_{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 1(iii)

_{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 21×0.5) 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 1(ii)

_{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 1(ii)

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 1(i)

_{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(i)

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 cm²Question 2

A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What is the volume of a packet containing 12 such matchboxes?Solution 2

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 cm³Question 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 m³ = 1000 litres.)Solution 3

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 4

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 4

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 5

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 5

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 6

_{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 6

Question 7

_{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 7

Question 8

_{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 8

_{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 + 6×2.5+ 2.5×8) m²}

_{= 2(48 + 15 + 20) m²}

_{= (2 x 83) m²=166 m²}Question 9

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 9

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. 5375Question 10

_{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 10

_LQuestion 11

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 11

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 12

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 12

Question 13

_{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 13

Question 14

_{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 14

Question 15

_{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 15

Question 16

_{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 16

Question 17In  shower, 5 cm of rain falls. Find the volume of water that falls on 2 hectares of ground.Solution 17

Question 18

_{Find the volume, the lateral surface area, the total surface area and the diagonal of cube, each of whose edges measures 9m. [Take ]}Solution 18

Question 19

_{The total surface area of a cube is 1176 cm². Find its volume.}Solution 19

Question 20

_{The lateral surface area of a cube is 900 cm². Find its volume.}Solution 20

Question 21

_{The volume of a cube is 512 cm³. Find its surface area.}Solution 21

Question 22

_{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 22

Question 23

_{Find the length of the longest pole that can be put in a room of dimensions (10mx 10m x5m).}Solution 23

Question 24

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 24

Question 25

Each edge of a cube is increased by 50%. Find the percentage increase in the surface area of the cube.Solution 25

Let the edge of the cube = ‘a’ cm

Then, surface area of cube = 6a² cm²

Question 26

If V is the volume of a cuboid of dimensions a, b, c and S is its surface area then prove that Solution 26

Question 27

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 27

Question 28

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 28

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

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

Exercise 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 cm}² _{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 cm}²_{. 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 litresQuestion 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,

Exercise Ex. 15D

Question 1(iii)

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

_{5 m}Solution 1(iii)

Question 1(ii)

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

_{4.2 cm}Solution 1(ii)

Question 1(i)

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

_{3.5 cm}Solution 1(i)

Question 2

_{The volume of a sphere is 38808 cm³. Find the radius and hence its surface area.}Solution 2

Question 3

_{Find the surface area of a sphere whose volume is 606.375 m³}Solution 3

Question 4

Find the volume of a sphere whose surface area is 154 cm².Solution 4

Surface area of sphere = 154 cm²

⇒ 4πr² = 154

Question 5

_{The surface area of a sphere is (576) cm². Find its volume.}Solution 5

Question 6

_{How many leads shots, each 3 mm in diameter, can be made from cuboid with dimensions  (12cm x 11cm x 9cm)?}Solution 6

Question 7

_{How many lead balls, each of radius 1 cm, can be made from a sphere of radius 8 cm?}Solution 7

Question 8

_{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 8

Question 9

_{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 9

Question 10

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

_{height 90 cm ?}Solution 10

Question 11

_{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 11

Question 12

_{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 12

Question 13

_{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 13

Question 14

_{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 14

Question 15

_{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 15

Question 16

The radii of two spheres are in the ratio 1:2. Find the ratio of their surface areas.Solution 16

Question 17

The surface areas of two spheres are in the ratio 1:4. Find the ratio of their volumes.Solution 17

Question 18

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 18

Question 19

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 waterSolution 19

Question 20

_{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 20

Question 21

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 21

Question 22

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 22

Question 23

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 23

Question 24

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 24

Question 25

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 25

Inner radius = 5 cm

⇒ Outer radius = 5 + 0.25 = 5.25 cm

Question 26

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 26

Inner diameter of the hemispherical bowl = 10.5 cm

Question 27

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 27

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 28

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of the hemisphere?Solution 28

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