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# Class 10 SELINA Solutions Maths Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise TEST YOURSELF

### Solution 1(a)

Correct option: (ii) 5pr2

Total surface area of the solid

= Curved surface area of cylinder + Curved surface area of cone + Base of the cylinder

= 2prh + prl + pr2

= 2pr(r) + pr(2r) + pr2

= 2pr2 + 2pr2 + pr2

= 5pr2

### Solution 1(b)

Correct option: (iii) 1 : 8

Let the radii of two solid spheres be r­1 and r2 respectively.

Then, r1 = 10 cm and r2 = 20 cm

The ratio between their volumes

### Solution 1(c)

Correct option: (iv) 20 cm

Let the radius and height of cone be r­1 and h respectively.

And, the radius of sphere be r2.

Then, h = 10 cm and r2 = 10 cm

Volume of cone = Volume of sphere

*The information provided in question is about cone and sphere and not about cylinder. Hence, as per information, the radius of cone is to be calculated and not of cylinder. Error in question.

### Solution 1(d)

Correct option: (i) 378

For a cuboid, l = 11 cm, b = 18 cm, h = 8 cm

For a sphere r = 1 cm

### Solution 1(e)

Correct option: (i) only 1

If a wire is extended four times along its length with same width all around, its volume will remain constant in both the cases.

But, the surface area of solid will increase. Hence, it will be not same in both the cases.

### Solution 2

Let the number of solid metallic spheres be 'n'

Volume of 1 sphere

Volume of metallic cone

=

The least number of spheres needed to form the cone is 15

### Solution 3

Radius of largest sphere that can be formed inside the cylinder should be equal to the radius of the cylinder.

Radius of the largest sphere = 7 cm

Volume of sphere

### Solution 4

Let the number of cones be 'n'.

Volume of the cylinder =

Volume of ice-cream cone = Volume of cone + Volume of hemisphere

Hence, number of cones required = 10

### Solution 5

Volume of the solid

### Solution 6

Diameter of a sphere = 6 cm

Diameter of cylindrical wire = 0.2 cm

Let length of wire = h

From (i) and (ii)

Hence, length of the wire = 36 m

### Solution 7

Let edge of the cube = a

volume of the cube =

The sphere, which exactly fits in the cube, has radius =

Therefore, volume of sphere =

Volume of cube : volume of sphere

### Solution 8

Radius of the base of poles (r) = 6 cm

Height of the cylindrical part (h1) = 110 cm

Height of the conical part (h2) = 9 cm

Total volume of the iron pole =

Weight of 1 cm3 = 8 gm

Therefore, total weight = 12780 8 = 102240 gm = 102.24 kg

### Solution 9

Side of square = 7 m

Length of the tunnel = 80 m

Area of cross section of the front part =

(i) Therefore, volume of the tunnel = area x length

(ii) Circumference of the front of tunnel

Therefore, surface area of the inner part of the tunnel

= 25 80

= 2000 m2

(iii) Area of floor = l b = 7 80 = 560 m2

### Solution 10

Diameter of cylindrical tank = 2.8 m

Height = 4.2 m

Volume of water filled in it =

Diameter of pipe = 7 cm

Let length of water in the pipe = h1

From (i) and (ii)

Therefore, time taken at the speed of 4 m per second

### Solution 11

Rate of flow of water = 9 km/hr

Water flow in 1 hour 15 minutes

i.e. in

Area of cross-section =

Therefore, volume of water

Dimensions of water tank = 7.5m × 5m × 4m

Area of tank = l × b = 7.5 × 5 = 37.5 m2

Let h be the height of water then,

37.5 × h = 28.125

### Solution 12

Diameter = 10 cm

Therefore, radius (r) = 5 cm

Height of the cone (h) = 12 cm

Height of the cylinder = 12 cm

(i) Total surface area of the solid

(ii) Total volume of the solid

(iii) Total weight of the solid = 1.7 kg

### Solution 13

Radius of cylinder = 3 cm

Height of cylinder = 6 cm

Radius of hemisphere = 2 cm

Height of cone = 4 cm

Volume of water in the cylinder when it is full =

Volume of water displaced = volume of cone + volume of

hemisphere

Therefore, volume of water which is left

### Solution 14

Radius of the cylinder = 3.5 m

Height = 7 m

(i) Total surface area of container excluding the base = Curved

surface area of the cylinder + area of hemisphere

(ii) Volume of the container =

### Solution 15

Total height of the tent = 85 m

Diameter of the base = 168 m

Therefore, radius (r) = 84 m

Height of the cylindrical part = 50 m

Then height of the conical part = (85 - 50) = 35 m

Slant height (l)

Total surface area of the tent =

Since 20% extra is needed for folds and stitching,

total area of canvas needed

### Solution 16

Volume of water filled in the test tube =

Volume of water filled up to 4 cm =

Let r be the radius and h be the height of test tube.

and

Dividing (i) by (ii)

Subtracting (ii) from (i)

Substituting the value of r in (iii)

Hence, Height = 20 cm and radius = 3.5 cm

### Solution 17

Diameter of hemisphere = 7 cm

Diameter of the base of the cone = 7 cm

Therefore, radius (r) = 3.5 cm

Height (h) = 8 cm

Volume of the solid =

Now, radius of cylindrical vessel (R) = 7 cm

Height (H) = 10 cm

Volume of water required to fill = 1540 - 192.5 = 1347.5 cm3

### Solution 18

Let the number of cones melted be n.

Let the radius of sphere be rs = 6 cm

Radius of cone be rc = 2 cm

And, height of the cone be h = 3 cm

Volume of sphere = n (Volume of a metallic cone)

### Solution 19

According to the condition in the question,

We know that,

l2 = r2 + h2

l2 = (7)2 + (24)2

l2 = 49 + 576

l2 = 625

l = 25 m

Curved Surface Area = πrl = × 7× 25 = 550m2

Therefore the height of the tent is 24m and it curved surface area is 550m2.

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise Ex. 20(A)

### Solution 2

Inner radius of pipe = 2.1 cm

Length of the pipe = 12 m = 1200 cm

### Solution 3

Depth of the well = 28 m

Therefore, volume of earth dug out =

Area of curved surface =

Cost of plastering at the rate of Rs 4.50 per sq m

= Rs 246.40 4.50

= Rs 1108.80

### Solution 4

External diameter of hollow cylinder = 20 cm

Therefore, external radius, R = 10 cm

Thickness = 0.25 cm

Hence, internal radius, r = (10 - 0.25) = 9.75 cm

Length of cylinder (h) = 15 cm

For solid cylinder,

Diameter = 2 cm

Therefore, radius (r) = 1 cm

Let h be the length

then,

Now, according to given condition:

Length of cylinder = 74.06 cm

### Solution 5

Diameter of the cylinder = 20 cm

Hence, Radius (r) = 10 cm

Height = h cm

(i) Curved surface area =

(ii) Volume of the cylinder =

or

### Solution 7

Diameter of cylindrical container = 42 cm

Therefore, radius (r) = 21 cm

Dimensions of rectangular solid = 22cm × 14cm × 10.5cm

Volume of solid =22 × 14 ×10.5 cm3 ...... (i)

Let height of water = h

Therefore, volume of water in the container =πr2

From (i) and (ii)

### Solution 8

Internal radius of the cylindrical container = 10 cm

Height of water = 7 cm

Therefore, surface area of the wetted surface =

### Solution 9

Length of an open pipe = 50 cm

External diameter = 20 cm ⇒ External radius (R) = 10 cm

Internal diameter = 6 cm ⇒ Internal radius (r) = 3 cm

Surface area of pipe open from both sides =

Area of upper and lower part =

Total surface area =4085.71+572=4657.71 cm2

### Solution 10

Ratio between height and radius of a cylinder = 3:1

Volume = …….(i)

Let radius of the base = r

then height = 3r

from (i) and (ii)

Therefore, radius = 7 cm and height = 3 x 7 = 21 cm

Now, total surface area =

### Solution 21

Total surface area of a hollow cylinder = 3575 cm2

Area of the base ring = 357.5 cm2

Height = 14 cm

Let thickness of the cylinder = d = (R-r)

Therefore, Total surface area =

and Area of base =

Dividing (i) by (ii)

Hence, thickness of the cylinder = 3.5 cm

### Solution 1(a)

Correct option: (iii) 4prh + 2pr2

Total surface area of open hollow cylinder with radius r and height h

= External curved surface area + Internal curved surface area + 2(Area of cross section)

= 2prh + 2prh + 2(pr2)

= 4prh + 2pr2

### Solution 1(b)

Correct option: (iii) S ÷ C

Curved surface area of a solid cylinder = Circumference of its base × Height

S = C × Height

Height = S ÷ C

### Solution 1(c)

Correct option: (iii)

Volume of cube submerged = Volume of water that rises

a3 = pr2h

### Solution 1(d)

Correct option: (ii)

Volume of big solid cylinder = pR2H

Volume of each small solid cylinder = pr2h

Then, number of smaller solid cylinders

### Solution 1(e)

Correct option: (iv) 2(pR2pr2) + 2pRh + 2prh

For an open pipe,

Length = H cm, External radius = R cm, Internal radius = r cm

Then, total surface area of an open pipe

= 2(Area of cross section) + External curved surface area + Internal curved surface area

= 2(pR2pr2) + 2pRh + 2prh

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise Ex. 20(B)

### Solution 2

Slant height () = 17 cm

But,

Now, volume of cone =

### Solution 3

Curved surface area =

Radius of base (r) = 56 cm

Let slant height =

Height of the cone =

### Solution 4

Circumference of the conical tent = 66 m

and height (h) = 12 m

Therefore, volume of air contained in it =

### Solution 5

The ratio between radius and height = 5:12

Volume = 5212 cubic cm

Let radius (r) = 5x, height (h) = 12x and slant height =

Now Volume =

### Solution 6

Let radius of each cone = r

Ratio between their slant heights = 5:4

Let slant height of the first cone = 5x

and slant height of second cone = 4x

Therefore, curved surface area of the first cone =

curved surface area of the second cone =

Hence, ratio between them =

### Solution 7

Let slant height of the first cone =

then slant height of the second cone = 2

Radius of the first cone =

Radius of the second cone =

Then, curved surface area of first cone =

curved surface area of second cone =

According to given condition:

### Solution 8

Diameter of the cone = 16.8 m

Therefore, radius (r) = 8.4 m

Height (h) = 3.5 m

(i) Volume of heap of wheat =

(ii) Slant height () =

Therefore, cloth required or curved surface area =

### Solution 9

Diameter of the tent = 48 m

Therefore, radius (r) = 24 m

Height (h) = 7 m

Slant height () =

Curved surface area =

Canvas required for stitching and folding

Total canvas required (area)

Length of canvas

Rate = Rs 24 per meter

Total cost

### Solution 10

Height of solid cone (h) = 8 cm

Volume of solid cone =

Height of smaller cone = 2 cm

Volume of smaller cone

Number of cones so formed

### Solution 11

Total surface area of cone =

slant height (l) = 13 cm

(i) Let r be its radius, then

Total surface area =

Either r+18 = 0, then r = -18 which is not possible

or r-5=0, then r = 5

(ii) Now

### Solution 13

Volume of vessel = volume of water =

diameter = 25.2 cm, therefore radius = 12.6 cm

height = 32 cm

Volume of water in the vessel =

On submerging six equal solid cones into it, one-fourth of the water overflows.

Therefore, volume of the equal solid cones submerged

= Volume of water that overflows

Now, volume of each cone submerged

### Solution 14

(i) Let r be the radius of the base of the conical tent, then area of the base floor =

Hence, radius of the base of the conical tent i.e. the floor = 7 m

(ii) Let h be the height of the conical tent, then the volume =

Hence, the height of the tent = 24 m

(iii) Let l be the slant height of the conical tent, then

The area of the canvas required to make the tent =

Length of the canvas required to cover the conical tent of its width 2 m =

### Solution 1(a)

Correct option: (iii)

For a cone, radius = height = r cm

Then, volume of the cone

### Solution 1(b)

Correct option: (ii)

For a cone, radius = r cm and height = h cm

Let the slant height = l cm

Then, its curved surface area = prl

### Solution 1(c)

Correct option: (iii) 140 cm

For a conical tent,

Height = 21 cm

Area of sheet required = Curved surface area of the tent-house

= prl

Let the smallest length of the paper = x cm

Then, x × 22 = 3080

- x = 140 cm

### Solution 1(d)

Correct option: (ii) 5.5 units

Let the radius of the cone = r units, height = h units and slant height = l units

Now,

### Solution 1(e)

Correct option: (ii) 7 : 4

Let the radius of each cone = r units

Let the slant height of two cones be l1 and l2 respectively.

The ratio of curved surface areas of two cones

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise Ex. 20(C)

### Solution 2

Volume of the sphere = 38808 cm3

Let radius of sphere = r

### Solution 3

Let the radius of spherical ball = r

Therefore, number of smaller balls made out of the given ball =

### Solution 4

Diameter of bigger ball = 8 cm

Therefore, Radius of bigger ball = 4 cm

Radius of small ball = 1 cm

Number of balls =

### Solution 5

Volume of first sphere = 27 volume of second sphere

Let radius of first sphere =

and radius of second sphere =

Therefore, volume of first sphere =

and volume of second sphere =

(i) Now, according to the question

(ii) Surface area of first sphere =

and surface area of second sphere =

Ratio in surface area =

### Solution 6

Let r be the radius of the sphere.

Surface area = 4πrand volume =

According to the condition:

Diameter of sphere = 2 × 3 cm = 6 cm

### Solution 7

Diameter of sphere =

Total curved surface area of each hemispheres =

### Solution 8

External radius (R) = 14 cm

(i) Internal curved surface area =

(ii) External curved surface area =

(iii) Total surface area =

(iv) Volume of material used =

### Solution 9

Let the radius of the sphere be 'r1'.

Let the radius of the hemisphere be 'r2'

TSA of sphere = 4∏r12

TSA of hemisphere = 3∏r22

TSA of sphere = TSA of hemi-sphere

Volume of sphere, V1 =

Volume of hemisphere, V2 =

Dividing V1 by V2,

### Solution 10

Let radius of the larger sphere be 'R'

Volume of single sphere

= Vol. of sphere 1 + Vol. of sphere 2 + Vol. of sphere 3

Surface area of the sphere

### Solution 1(a)

Correct option: (iii) 512

Radius of big sphere, R = 16 cm

Radius of each small sphere, r = 2 cm

Then, number of spheres formed

### Solution 1(b)

Correct option: (ii) 3pR2 pr2

Outer surface area of the bowl

= Total surface area of hemispherical bowl – pr2

= 3pR2 pr2

### Solution 1(c)

Correct option: (iv) 9 : 4

Let the radii of two spherical solids be r1 and r2 respectively.

Then,

Now, ratio of their curved surface areas

### Solution 1(d)

Correct option: (i) 4 cm

Let the diameter of each small sphere = 2r cm

Then, radius of each small sphere = r cm

Now, Volume of big sphere = Volume of 64 small spheres

Therefore, diameter = 2r = 4 cm

### Solution 1(e)

Correct option: (iv)

Let the radius of big sphere formed = R

Now, Volume of big sphere = Volume of three small spheres

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise Ex. 20(D)

### Solution 3

External diameter = 8 cm

Therefore, radius (R) = 4 cm

Internal diameter = 4 cm

Therefore, radius (r) = 2 cm

Volume of metal used in hollow sphere =

Diameter of cone = 8 cm

Let height of cone = h

From (i) and (ii)

Height of the cone = 14 cm

### Solution 4

Volume of spherical shell

Volume of solid circular cone

Vol. of Cone = Vol. of sphere

Hence, diameter = 2r = 7 cm

### Solution 5

Let the radius of the smaller cone be 'r' cm.

Volume of larger cone

Volume of smaller cone

Volume of larger cone=3× Volume of smaller cone

### Solution 6

Volume of rectangular block =

Let r be the radius of sphere

From (i) and (ii)

Radius of sphere = 21 cm

### Solution 7

Radius of hemispherical bowl = 9 cm

Diameter each of cylindrical bottle = 3 cm

Radius = cm, and height = 4 cm

### Solution 8

Total area of solid metallic sphere = 1256 cm2

(i)Let radius of the sphere is r then

(ii) Volume of sphere = Volume of right circular cone =

Number of cones

### Solution 1(a)

Correct option: (ii) 40 cm

Let the height of cone = h cm

Volume of cone = Volume of sphere

### Solution 1(b)

Correct option: (iv) 1 : 4

Volume of cone = Volume of sphere

### Solution 1(c)

Correct option: (iii) 8 : 1

For a cone, radius : height = r : h = 2 : 1

### Solution 1(d)

Correct option: (i) 3

### Solution 1(e)

Correct option: (ii) 4 × p m

For a cone, radius, r = 10 cm and height, h = 12 cm

Volume of cone

Volume of cylinder = Volume of cone

Area of cross-section × length = Volume of cone

1 × L = 400p

L = 400 × p cm = 4 × p m

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise Ex. 20(E)

### Solution 2

Height of cone = 15 cm

and radius of the base = cm

Therefore, volume of the solid = volume of the conical part + volume of hemispherical part.

### Solution 3

Radius of hemispherical part (r) = 3.5 m =

Therefore, Volume of hemisphere =

Volume of conical part = (2/3 of hemisphere)

Let height of the cone = h

Then,

Height of the cone = 4.67 m

Surface area of buoy =

But

Therefore, Surface area =

Surface Area = 141.17 m2

### Solution 4

(i) Total surface area of cuboid = 2(b + bh + h)

=2(42 × 30 + 30 × 20 + 20 × 42)

=2(1260 + 600 + 840)

=2 × 2700

=5400 cm2

Diameter of the cone = 14 cm

⇒ Radius of the cone =

Area of circular base

Area of curved surface area of cone

Surface area of remaining part = 5400 + 550 - 154 =5796 cm2

(ii) Dimensions of rectangular solids = (42 × 30 × 20) cm

volume = (42 × 30 × 20) = 25200 cm3

Radius of conical cavity (r) =7 cm

height (h) = 24 cm

Volume of cone =

Volume of remaining solid = (25200 - 1232) = 23968 cm3

(iii) Weight of material drilled out

=1232 × 7 g = 8624g = 8.624 kg

### Solution 5

The diameter of the largest hemisphere that can be placed on a face of a cube of side 7 cm will be 7 cm.

Its curved surface area =

Surface area of the top of the resulting solid = Surface area of the top face of the cube - Area of the base of the hemisphere

Surface area of the cube = 5 × (side)2 = 5 × 49 = 245 cm2 .........(iii)

Total area of resulting solid = 245 + 10.5 + 77 = 332.5 cm2

### Solution 6

Height of cone = 8 cm

Volume =

Therefore, volume of water that flowed out =

Radius of each ball = 0.5 cm =

Volume of a ball =

Therefore, No. of balls =

Hence, number of lead balls = 100

### Solution 7

Let r be the radius of the bowl.

Capacity of the bowl =

### Solution 8

For the volume of cone to be largest, h = r cm

Volume of the cone

### Solution 9

Let the height of the solid cones be 'h'

Volume of solid circular cones

Volume of sphere

Volume of sphere = Volume of cone 1 + volume of cone 2

### Solution 10

Let the radius of base be 'r' and the height be 'h'

Volume of cone, Vc

Volume of hemisphere, Vh

### Solution 1(a)

Correct option: The given options are not applicable.

Volume of whole body = Volume of cone + Volume of hemisphere

### Solution 1(b)

Correct option: (iii) 2p cm3

Volume of the body = Volume of cone + Volume of cylinder

### Solution 1(c)

Correct option: (ii) pr2 + prl + 2prh

Total surface area of the remaining solid

= Base of the cylinder + Curved surface area of cone + Curved surface area of cylinder

= pr2 + prl + 2prh

### Solution 1(d)

Correct option: (i) 2prh + pr2 + prl

Wetted surface area of the whole body

= Curved surface area of cylinder + Base of the cylinder + Curved surface area of cone

= 2prh + pr2 + prl

### Solution 1(e)

Correct option: (iii)

Volume of air left in the cylinder

= Volume of cylinder – Volume of sphere

## Cylinder, Cone and Sphere (Surface Area and Volume) Exercise Ex. 20(F)

### Solution 2

Height of the cylinder (h) = 10 cm

and radius of the base (r) = 6 cm

Volume of the cylinder =

Height of the cone = 10 cm

Radius of the base of cone = 6 cm

Volume of the cone =

Volume of the remaining part

### Solution 3

Radius of solid cylinder (R) = 12 cm

and Height (H) = 16 cm

Radius of cone (r) = 6 cm, and height (h) = 8 cm.

(i) Volume of remaining solid

(ii) Slant height of cone

Therefore, total surface area of remaining solid = curved surface area of cylinder + curved surface area of cone + base area of cylinder + area of circular ring on upper side of cylinder

### Solution 4

Radius of the cylindrical part of the tent (r) =

Slant height () = 80 m

Therefore, total curved surface area of the tent =

Width of canvas used = 1.5 m

Length of canvas =

Total cost of canvas at the rate of Rs 15 per meter

### Solution 5

Height of the cylindrical part = H = 8 m

Height of the conical part = h = (13-8)m = 5 m

Diameter = 24 m radius = r = 12 m

Slant height of the cone = l

Slant height of cone = 13 m

(i) Total surface area of the tent =

(ii)Area of canvas used in stitching = total area

### Solution 6

Diameter of cylindrical container = 42 cm

Therefore, radius (r) = 21 cm

Dimensions of rectangular solid = 22cm 14cm 10.5cm

Volume of solid =

Let height of water = h

Therefore, volume of water in the container =

From (i) and (ii)

### Solution 7

Diameter of spherical marble = 1.4 cm

Volume of one ball

Diameter of beaker = 7 cm

Height of water = 5.6 cm

Volume of water =

No. of balls dropped

### Solution 8

Length = 21 cm, Breadth = 7 cm

Area of cross section of the water channel =

Flow of water in one minute at the rate of 20 cm per second

Length of the water column = 20 60 = 1200 cm

Therefore, volume of water =

### Solution 9

Diameter of the base of the cylinder = 7 cm

Therefore, radius of the cylinder =

Volume of the cylinder

Diameter of the base of the cone =

Therefore, radius of the cone =

Volume of the cone

On placing the cone into the cylindrical vessel, the volume of the remaining portion where the water is to be filled

Height of new cone =

Therefore, volume of new cone

Volume of water which comes down =

Let h be the height of water which is dropped down.

From (i) and (ii)

Drop in water level =

### Solution 10

Radius of the base of the cylindrical can = 3.5 cm

(i) When the sphere is in can, then total surface area of the can =

Base area + curved surface area

(ii) Let depth of water = x cm

When sphere is not in the can, then volume of the can =

volume of water + volume of sphere

### Solution 11

Let the height of the water level be 'h', after the solid is turned upside down.

Volume of water in the cylinder

Volume of the hemisphere

Volume of water in the cylinder

= Volume of water level - Volume of the hemisphere

### Solution 1(a)

Correct option: (iii) 1 : 3

Let the heights of cylinder and cone be H and h.

Volume of cylinder = Volume of cone

### Solution 1(b)

Correct option: (iii) 2pr3

Volume of the given solid

= Volume of hemisphere + Volume of cylinder + Volume of cone

### Solution 1(c)

Correct option: (ii) 2 cm

Let the radius of the solid sphere formed = R cm

Volume of 8 identical spheres = Volume of solid sphere formed

### Solution 1(d)

Correct option: (i) 27 : 32

Let the radii of two cones be r­1 and r2 respectively.

And, let their heights be h­1 and h2 respectively.

Now, the ratio between their volumes

### Solution 1(e)

Correct option: (iv) 300%

Percentage increase in volume