RD Sharma Solutions Class 9 Maths Chapter 21 Surface Area and Volume of Sphere

RD Sharma Solutions Class 9 Maths Chapter 21 – Free PDF Download

RD Sharma Solutions for Class 9 Maths Chapter 21 Surface Area and Volume of a Sphere are given here, which consists of questions and answers related to spheres. These questions are really helpful for the preparation of annual exams as well as other competitive exams. Here, all the textbook questions in the RD Sharma Solutions for Class 9 are given in a detailed and step-by-step format to help the students understand the concepts and clear all their doubts easily.

Students can find several exercises in Class 9 Maths Chapter 21 – Surface Area and Volume of Sphere. Click on the links below to download the RD Sharma Solutions of this chapter for in-depth learning of concepts. In this chapter, students will learn to find the surface area and volume of the sphere. Let us have a brief look at what is a sphere in three-dimensional space: they are a set of points at an equal distance ‘r’ from a given point. The shape resembles a ball.

RD Sharma Solutions for Class 9 Maths Chapter 21 Surface Area and Volume of a Sphere

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Access Answers to RD Sharma Solutions for Class 9 Maths Chapter 21 Surface Area and Volume of a Sphere

Exercise 21.1 Page No: 21.8

Question 1: Find the surface area of a sphere of radius:

(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm

Solution:

The surface area of a sphere = 4πr²

Where, r = radius of a sphere

(i) Radius = 10.5 cm

Surface area = 4 x 22/7 x (10.5)²

= 1386

The surface area is 1386 cm²

(ii) Radius= 5.6 cm

Surface area = 4×22/7×(5.6)²

= 394.24

The surface area is 394.24 cm²

(iii) Radius = 14 cm

Surface area = 4×22/7×(14)²

= 2464

The surface area is 2464 cm²

Question 2: Find the surface area of a sphere of diameter:

(i) 14 cm (ii) 21 cm (iii) 3.5 cm

Solution:

The surface area of a sphere = 4πr²

Where, r = radius of a sphere

(i) Diameter= 14 cm

So, Radius = Diameter/2 = 14/2 cm = 7 cm

Surface area = 4×22/7×(7)²

= 616

The surface area is 616 cm²

(ii) Diameter = 21cm

So, Radius = Diameter/2 = 21/2 cm = 10.5 cm

Surface area= 4×22/7×(10.5)²

= 1386

The surface area is 1386 cm²

(iii) Diameter= 3.5cm

So, Radius = Diameter/2 = 3.5/2 cm = 1.75 cm

Surface area = 4×22/7×(1.75)²

= 38.5

The surface area is 38.5 cm²

Question 3: Find the total surface area of a hemisphere and a solid hemisphere, each of radius 10 cm. (π=3.14)

Solution:

Radius of a hemisphere = Radius of a solid hemisphere = 10 cm (Given)

The surface area of the hemisphere = 2πr²

= 2×3.14×(10)² cm²

= 628 cm²

And, the surface area of the solid hemisphere = 3πr²

= 3×3.14×(10)² cm²

= 942 cm²

Question 4: The surface area of a sphere is 5544 cm², find its diameter.

Solution:

The surface area of a sphere is 5544 cm²

The surface area of a sphere = 4πr²

So, 4πr² = 5544

4×22/7×(r)² = 5544

r² = (5544 × 7)/88

r² = 441

or r = 21cm

Now, Diameter=2(radius) = 2(21) = 42cm

Question 5: A hemispherical bowl made of brass has an inner diameter 10.5 cm. Find the cost of tin plating it on the inside at the rate of Rs.4 per 100 cm².

Solution:

The inner diameter of the hemispherical bowl = 10.5 cm

So, radius = Diameter/2 = 10.5/2 cm = 5.25 cm

Now, the surface area of the hemispherical bowl = 2πr²

= 2 × 3.14 × (5.25)²

= 173.25

So, the surface area of the hemispherical bowl is 173.25 cm²

Find the cost:

Cost of tin plating 100 cm² area= Rs.4 (given)

Cost of tin plating 173.25cm² area = Rs. 4×173.25100 = Rs. 6.93

Therefore, the cost of tin plating the inner side of the hemispherical bowl is Rs. 6.93.

Question 6: The dome of a building is in the form of a hemisphere. Its radius is 63 dm. Find the cost of painting it at the rate of Rs. 2 per sq m.

Solution:

The radius of hemispherical dome = 63 dm or 6.3 m

The inner surface area of dome = 2πr²

=2×3.14×(6.3)²

= 249.48

So, the inner surface area of the dome is 249.48 m²

Now find the cost:

Cost of painting 1m² = Rs.2 (given)

Therefore, the cost of painting 249.48 m²= Rs. (249.48×2) = Rs.498.96.

Exercise 21.2 Page No: 21.19

Question 1: Find the volume of a sphere whose radius is:

(i) 2 cm (ii) 3.5 cm (iii) 10.5 cm.

Solution:

The volume of a sphere = 4/3πr³ Cubic Units

Where, r = radius of a sphere

(i) Radius = 2 cm

Volume = 4/3 × 22/7 × (2)³

= 33.52

Volume = 33.52 cm³

(ii) Radius = 3.5cm

Therefore volume = 4/3×22/7×(3.5)³

= 179.666

Volume = 179.666 cm³

(iii) Radius = 10.5 cm

Volume = 4/3×22/7×(10.5)³

= 4851

Volume = 4851 cm³

Question 2: Find the volume of a sphere whose diameter is:

(i) 14 cm (ii) 3.5 dm (iii) 2.1 m

Solution:

The volume of a sphere = 4/3πr³ Cubic Units

Where, r = radius of a sphere

(i) diameter =14 cm

So, radius = diameter/2 = 14/2 = 7cm

Volume = 4/3×22/7×(7)³

= 1437.33

Volume = 1437.33 cm³

(ii) diameter = 3.5 dm

So, radius = diameter/2 = 3.5/2 = 1.75 dm

Volume = 4/3×22/7×(1.75)³

= 22.46

Volume = 22.46 dm³

(iii) diameter = 2.1 m

So, radius = diameter/2 = 2.1/2 = 1.05 m

Volume = 4/3×22/7×(1.05)³

= 4.851

Volume = 4.851 m³

Question 3: A hemispherical tank has an inner radius of 2.8 m. Find its capacity in litres.

Solution:

The radius of hemispherical tank = 2.8 m

Capacity of hemispherical tank = 2/3 πr³

=2/3×22/7×(2.8)³ m³

= 45.997 m³

Therefore, capacity in litres = 45997 litres

Question 4: A hemispherical bowl is made of steel 0.25 cm thick. The inside radius of the bowl is 5 cm. Find the volume of steel used in making the bowl.

Solution:

The inner radius of a hemispherical bowl = 5 cm

Outer radius of a hemispherical bowl = 5 cm + 0.25 cm = 5.25 cm

The volume of steel used = Outer volume – Inner volume

= 2/3×π×((5.25)³−(5)³)

= 2/3×22/7×((5.25)³−(5)³)

= 41.282

The volume of steel used is 41.282 cm³

Question 5: How many bullets can be made out of a cube of lead, whose edge measures 22 cm, each bullet being 2 cm in diameter?

Solution:

Edge of a cube = 22 cm

Diameter of bullet = 2 cm

So, the radius of the bullet (r) = 1 cm

Volume of the cube = (side)³ = (22)³ cm³ = 10648 cm³

And,

The volume of each bullet which will be spherical in shape = 4/3πr³

= 4/3 × 22/7 × (1)³ cm³

= 4/3 × 22/7 cm³

= 88/21 cm³

Number of bullets = (Volume of cube) / (Volume of bullet)

= 10648/88/21

= 2541

Therefore, 2541 bullets can be made.

Question 6: A shopkeeper has one laddoo of radius 5 cm. With the same material, how many laddoos of radius 2.5 cm can be made?

Solution:

The volume of laddoo having a radius of 5 cm (V1) = 4/3×22/7×(5)³

= 11000/21 cm³

Also, the volume of the laddoo having a radius of 2.5 cm (V2) = 4/3πr³

= 4/3×22/7×(2.5)³ cm³

= 1375/21 cm³

Therefore,

Number of laddoos of radius 2.5 cm that can be made = V1/V2 = 11000/1375 = 8

Question 7: A spherical ball of lead 3 cm in diameter is melted and recast into three spherical balls. If the diameters of two balls be 3/2cm and 2 cm, find the diameter of the third ball.

Solution:

The volume of the lead ball with a radius of 3/2 cm = 4/3πr³

= 4/3×π×(3/2)³

Let, Diameter of the first ball (d1) = 3/2cm

The radius of the first ball (r1) = 3/4 cm

Diameter of the second ball (d2) = 2 cm

Radius of second ball (r2) = 2/2 cm = 1 cm

Diameter of the third ball (d3) = d

The radius of the third ball (r3) = d/2 cm

Now,

So, the diameter of the third ball is 2.5 cm.

Question 8: A sphere of radius 5 cm is immersed in water filled in a cylinder, and the level of water rises 5/3 cm. Find the radius of the cylinder.

Solution:

Radius of sphere = 5 cm (Given)

Let ‘r’ be the radius of the cylinder.

We know, the volume of the sphere = 4/3πr³

By putting values, we get

= 4/3×π×(5)³

The height (h) of water rise is 5/3 cm (Given)

The volume of water rise in the cylinder = πr²h

Therefore, the volume of water rise in the cylinder = Volume of the sphere

So, πr²h = 4/3πr³

π r² × 5/3 = 4/3 × π × (5)³

or r² = 100

or r = 10

Therefore, the radius of the cylinder is 10 cm.

Question 9: If the radius of a sphere is doubled, what is the ratio of the volume of the first sphere to that of the second sphere?

Solution:

Let r be the radius of the first sphere, then 2r be the radius of the second sphere.

Now,

The ratio of the volume of the first sphere to the second sphere is 1:8.

Question 10: A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights.

Solution:

The volume of the cone = Volume of the hemisphere (Given)

1/3πr²h = 2/3 πr³

(Using respective formulas)

r²h = 2r³

or h = 2r

Since a cone and a hemisphere have equal bases, it implies they have the same radius.

h/r = 2

or h : r = 2 : 1

Therefore, the ratio of their heights is 2:1

Question 11: A vessel in the form of a hemispherical bowl is full of water. Its contents are emptied in a right circular cylinder. The internal radii of the bowl and the cylinder are 3.5 cm and 7 cm, respectively. Find the height to which the water will rise in the cylinder.

Solution:

The volume of water in the hemispherical bowl = Volume of water in the cylinder … (Given)

The inner radius of the bowl ( r₁) = 3.5cm

The inner radius of cylinder (r₂) = 7cm

The volume of water in the hemispherical bowl = Volume of water in the cylinder

2/3πr₁³ = πr₂²h

Where h is the height to which water rises in the cylinder.

2/3π(3.5)³ = π(7)²h

or h = 7/12

Therefore, 7/12 cm is the height to which water rises in the cylinder.

Question 12: A cylinder whose height is two thirds of its diameter, has the same volume as a sphere of radius 4 cm. Calculate the radius of the base of the cylinder.

Solution:

Radius of a sphere (R)= 4 cm (Given)

Height of the cylinder = 2/3 diameter (given)

We know, Diameter = 2(Radius)

Let h be the height and r be the base radius of a cylinder, then

h = 2/3× (2r) = 4r/3

The volume of the cylinder = Volume of the sphere

πr²h = 4/3πR³

π × r² × (4r/3) = 4/3 π (4)³

(r)³ = (4)³

or r = 4

Therefore, the radius of the base of the cylinder is 4 cm.

Question 13: A vessel in the form of a hemispherical bowl is full of water. The contents are emptied into a cylinder. The internal radii of the bowl and cylinder are respectively 6 cm and 4 cm. Find the height of water in the cylinder.

Solution:

Radius of a bowl (R)= 6 cm (Given)

Radius of a cylinder (r) = 4 cm (given)

Let h be the height of a cylinder.

Now,

The volume of water in the hemispherical bowl = Volume of the cylinder

2/3 π R³ = πr² h

2/3 π (6)³ = π(4)² h

or h = 9

Therefore, the height of the water in the cylinder is 9 cm.

Question 14: A cylindrical tub of radius 16 cm contains water to a depth of 30 cm. A spherical iron ball is dropped into the tub, and thus the level of water is raised by 9 cm. What is the radius of the ball?

Solution:

Let r be the radius of the iron ball.

Radius of the cylinder (R) = 16 cm (Given)

A spherical iron ball is dropped into the cylinder, and thus the level of water is raised by 9 cm. So, height (h) = 9 cm

From statement,

The volume of iron ball = Volume of water raised in the hub

4/3πr³ = πR²h

4/3 r³ = (16)² × 9

or r³ = 1728

or r = 12

Therefore, radius of the ball = 12cm.

Exercise VSAQs Page No: 21.25

Question 1: Find the surface area of a sphere of radius 14 cm.

Solution:

The radius of a sphere (r) = 14 cm

The surface area of a sphere = 4πr²

= 4 × (22/7) × 14² cm²

= 2464 cm²

Question 2: Find the total surface area of a hemisphere of radius 10 cm.

Solution:

The radius of a hemisphere (r) = 10 cm

The total surface area of a hemisphere = 3πr²

= 3 × (22/7) × 10² cm²

= 942 cm²

Question 3: Find the radius of a sphere whose surface area is 154 cm².

Solution:

The surface area of a sphere = 154 cm²

We know, Surface area of a sphere = 4πr²

So, 4πr² = 154

4 x 22/7 x r² = 154

r² = 49/4

or r = 7/2 = 3.5

The radius of a sphere is 3.5 cm.

Question 4: The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding.

Solution:

Diameter of hollow sphere = 7 m

So, radius of hollow sphere = 7/2 m = 3.5 cm

Now,

The area available to the motorcyclist for riding = Surface area of a sphere = 4πr²

= 4 × (22/7) × 3.5² m²

= 154 m²

Question 5: Find the volume of a sphere whose surface area is 154 cm².

Solution:

The surface area of a sphere = 154 cm²

We know, Surface area of a sphere = 4πr²

So, 4πr² = 154

4 x 22/7 x r² = 154

or r² = 49/4

or r = 7/2 = 3.5

Radius (r) = 3.5 cm

Now,

Volume of sphere = 4/3 π r³

= (4/3) π × 3.5³

= 179.66

Therefore, the volume of the sphere is 179.66 cm³.

RD Sharma Solutions for Class 9 Maths Chapter 21 Surface Area and Volume of a Sphere

In Chapter 21 of Class 9 RD Sharma Solutions, students will study important concepts as listed below:

• Sphere introduction
• Section of a sphere by a plane
• Surface area of a sphere, hemisphere and spherical shell
• Volume of a sphere, hemisphere and spherical shell