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

Question 1

The surface area of a sphere is 2464 cm², find its volume.

Solution 1

Surface area of the sphere = 2464 cm²

Let radius = r, then

Volume = 

Question 2

The volume of a sphere is 38808 cm³; find its diameter and the surface area.

Solution 2

Volume of the sphere = 38808 cm³

Let radius of sphere = r

Question 3

A spherical ball of lead has been melted and made into identical smaller balls with radius equal to half the radius of the original one. How many such balls can be made?

Solution 3

Let the radius of spherical ball = r

Radius of smaller ball = 

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

Question 4

How many balls each of radius 1 cm can be made by melting a bigger ball whose diameter is 8 cm.

Solution 4

Diameter of bigger ball = 8 cm

Therefore, Radius of bigger ball = 4 cm

Radius of small ball = 1 cm

Number of balls = 

Question 5

8 metallic sphere; each of radius 2 mm, are melted and cast into a single sphere. Calculate the radius of the new sphere.

Solution 5

Radius of metallic sphere = 2 mm = 

Volume of 8 spheres = 

Let radius of new sphere = R

From (i) and (ii)

Question 6

The volume of one sphere is 27 times that of another sphere. Calculate the ratio of their:

(i) radii

(ii) surface areas

Solution 6

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 = 

Question 7

If the number of square centimeters on the surface of a sphere is equal to the number of cubic centimeters in the volume, what is the diameter of the sphere?

Solution 7

Let r be the radius of the sphere.

Surface area =  and volume = 

According to the condition:

Diameter of sphere = 2  3 cm = 6 cm

Question 8

A solid metal sphere is cut through its centre into 2 equal parts. If the diameter of the sphere is find the total surface area of each part correct to 2 decimal places.

Solution 8

Diameter of sphere = 

Therefore, radius of sphere = 

Total curved surface area of each hemispheres =

Question 9

The internal and external diameters of a hollow hemi-spherical vessel are 21 cm and 28 cm respectively. Find:

(i) internal curved surface area

(ii) external curved surface area

(iii) total surface area

(iv) volume of material of the vessel.

Solution 9

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 =

Question 10

A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.

Solution 10

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,

 

 

Question 11

Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively are melted and recasted into a single solid sphere. Taking ∏ = 3.1, find the surface area of solid sphere formed.

Solution 11

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

 

 

 

 

Question 12

The surface area of a solid sphere is increased by 21% without changing its shape. Find the percentage increase in its:

(i) radius

(ii) volume

Solution 12

Let the radius of the sphere be 'r'.

Total surface area the sphere, S  

New surface area of the sphere, S'

 

(i) Let the new radius be r₁

 

Percentage change in radius

Percentage change in radius = 10%

(ii) Let the volume of the sphere be V

Let the new volume of the sphere be V'.

Percentage change in volume =33.1%