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

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

Correct option: (iii) 1 : 8

Let the radii of two solid spheres be r_­1 and r₂ respectively.

Then, r₁ = 10 cm and r₂ = 20 cm

The ratio between their volumes

Correct option: (iv) 20 cm

Let the radius and height of cone be r_­1 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.

Correct option: (i) 378

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

For a sphere r = 1 cm

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.

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

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

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

 Volume of the solid

Diameter of a sphere = 6 cm

Radius = 3 cm

Diameter of cylindrical wire = 0.2 cm

Therefore, radius of wire =

Let length of wire = h

From (i) and (ii)

Hence, length of the wire = 36 m

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

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

Side of square = 7 m

Radius of semicircle =

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

Diameter of cylindrical tank = 2.8 m

Therefore, radius = 1.4 m

Height = 4.2 m

Volume of water filled in it =

Diameter of pipe = 7 cm

Therefore, radius (r) =

Let length of water in the pipe = h₁

From (i) and (ii)

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

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

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

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

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 =

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

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

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

Let the number of cones melted be n.

Let the radius of sphere be r_s = 6 cm

Radius of cone be r_c = 2 cm

And, height of the cone be h = 3 cm

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

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. 

Inner radius of pipe = 2.1 cm

Length of the pipe = 12 m = 1200 cm

Radius of the well =

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

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

Diameter of the cylinder = 20 cm

Hence, Radius (r) = 10 cm

Height = h cm

(i) Curved surface area =

(ii) Volume of the cylinder =

or

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 =πr²h  

From (i) and (ii)

Internal radius of the cylindrical container = 10 cm

Height of water = 7 cm

Therefore, surface area of the wetted surface =

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 

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 =

Total surface area of a hollow cylinder = 3575 cm²

Area of the base ring = 357.5 cm²

Height = 14 cm

Let external radius = R and internal radius = r

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

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

Correct option: (iii) S ÷ C

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

⇒ S = C × Height

⇒ Height = S ÷ C

Correct option: (iii)

Volume of cube submerged = Volume of water that rises

⇒ a³ = pr²h

Correct option: (ii)

Volume of big solid cylinder = pR²H

Volume of each small solid cylinder = pr²h

Then, number of smaller solid cylinders

Correct option: (iv) 2(pR² – pr²) + 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(pR² – pr²) + 2pRh + 2prh

Slant height (ℓ) = 17 cm

Radius (r) = 8 cm

But,

Now, volume of cone =

Curved surface area =

Radius of base (r) = 56 cm

Let slant height =

Height of the cone =

Circumference of the conical tent = 66 m

and height (h) = 12 m

Therefore, volume of air contained in it =

The ratio between radius and height = 5:12

Volume = 5212 cubic cm

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

Now Volume =

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 =

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:

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 =

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

Height of solid cone (h) = 8 cm

Radius (r) = 6 cm

Volume of solid cone =

Height of smaller cone = 2 cm

and radius =

Volume of smaller cone

Number of cones so formed

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

Therefore, radius = 5 cm

(ii) Now

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

(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 =

Correct option: (iii)

For a cone, radius = height = r cm

Then, volume of the cone

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

Correct option: (iii) 140 cm

For a conical tent,

Radius = 28 cm

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

Correct option: (ii) 5.5 units

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

Now,

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

Volume of the sphere = 38808 cm³

Let radius of sphere = r

Let the radius of spherical ball = r

Radius of smaller ball =

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

Diameter of bigger ball = 8 cm

Therefore, Radius of bigger ball = 4 cm

Radius of small ball = 1 cm

Number of balls =

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 =

Let r be the radius of the sphere.

Surface area = 4πr² and volume =

According to the condition:

Diameter of sphere = 2 × 3 cm = 6 cm

Diameter of sphere =

Therefore, radius of sphere =

Total curved surface area of each hemispheres =

External radius (R) = 14 cm

Internal radius (r) =

(i) Internal curved surface area =

(ii) External curved surface area =

(iii) Total surface area =

(iv) Volume of material used =

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,

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

Correct option: (iii) 512

Radius of big sphere, R = 16 cm

Radius of each small sphere, r = 2 cm

Then, number of spheres formed

Correct option: (ii) 3pR² – pr²

Outer surface area of the bowl

= Total surface area of hemispherical bowl – pr²

= 3pR² – pr²

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

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

Correct option: (iv)

Let the radius of big sphere formed = R

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

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

Therefore, radius = 4 cm

Let height of cone = h

From (i) and (ii)

Height of the cone = 14 cm

Internal radius = 3cm

External radius = 5 cm

Volume of spherical shell

Volume of solid circular cone

Vol. of Cone = Vol. of sphere

Hence, diameter = 2r = 7 cm

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

Volume of rectangular block =

Let r be the radius of sphere

From (i) and (ii)

Radius of sphere = 21 cm

Radius of hemispherical bowl = 9 cm

Diameter each of cylindrical bottle = 3 cm

Radius = cm, and height = 4 cm

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

Correct option: (ii) 40 cm

Let the height of cone = h cm

Volume of cone = Volume of sphere

Correct option: (iv) 1 : 4

Volume of cone = Volume of sphere

Correct option: (iii) 8 : 1

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

Correct option: (i) 3

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

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.

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

(i) Total surface area of cuboid = 2(ℓb + bh + ℓh)

=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

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

Therefore, radius =

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

Height of cone = 8 cm

Radius = 5 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

Let r be the radius of the bowl.

Capacity of the bowl =

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

Volume of the cone

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

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

Volume of cone, V_c

Volume of hemisphere, V_h

Correct option: The given options are not applicable.

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

Correct option: (iii) 2p cm³

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

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

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

Correct option: (iii)

Volume of air left in the cylinder

= Volume of cylinder – Volume of sphere

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

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

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

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

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)

Diameter of spherical marble = 1.4 cm

Therefore, radius = 0.7 cm

Volume of one ball

Diameter of beaker = 7 cm

Therefore, radius =

Height of water = 5.6 cm

Volume of water =

No. of balls dropped

Length = 21 cm, Breadth = 7 cm

Radius of semicircle =

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 =

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 =

Radius = 2 cm

Therefore, volume of new cone

Volume of water which comes down =

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

Radius =

From (i) and (ii)

Drop in water level =

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

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

Correct option: (iii) 1 : 3

Radius of cylinder = Radius of cone = r

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

Volume of cylinder = Volume of cone

Correct option: (iii) 2pr³

Volume of the given solid

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

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

Correct option: (i) 27 : 32

Let the radii of two cones be r_­1 and r₂ respectively.

And, let their heights be h_­1 and h₂ respectively.

Now, the ratio between their volumes

Correct option: (iv) 300%

Percentage increase in volume