RD Sharma Solutions for Class 9 Maths Chapter 19 Surface Area and Volume of A Right Circular Cylinder

RD Sharma Solutions Class 9 Chapter 19 is about right circular cylinder which is a 3D geometrical structure having 2 circular base and 2 parallel faces. In this chapter, students will learn how to find surface area and volume of a cylinder.

Here is the formula which is applied to calculate the surface area and volume of a circular cylinder.

Surface area of a cylinder = 2πr (r + h) and

Volume of a cylinder = πr²h

Where r = radius of the cylinder and h = height of the cylinder.

R D Sharma solutions for class 9 Chapter 19 are given here so that students can prepare for their exam more effectively.

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Exercise 19.1 Page No: 19.7

Question 1: Curved surface area of a right circular cylinder is 4.4 m². If the radius of the base of the cylinder is 0.7 m. Find its height.

Solution:

Radius of the base of the cylinder = r = 0.7 m (Given)

Curved surface area of cylinder = C.S.A = 4.4m² (Given)

Let ‘h’ be the height of the cylinder.

We know, curved surface area of a cylinder = 2πrh

Therefore,

2πrh = 4.4

2 x 3.14 x 0.7 x h = 4.4

or h = 1

Therefore the height of the cylinder is 1 m.

Question 2: In a hot 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:

Height of cylinder (h) = Length of cylindrical pipe = 28 m or 2800 cm (Given)

Diameter of circular end of pipe = 5 cm (given)

Let ‘r’ be the radius of circular end, then r = diameter/2 = 5/2 cm

We know, Curved surface area of cylindrical pipe = 2πrh

= 2 x 3.14 x 5/2 x 2800

= 44000

Therefore, the area of radiating surface is 44000 cm².

Question 3: A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs 12.50 per m².

Solution:

Height of cylindrical pillar (h) = 3.5 m

Radius of circular end of pillar ( r) = 50/2 cm = 25 cm = 0.25 m

Curved surface area of cylindrical pillar = 2πrh

= 2 x 3.14 x 0.25 x 3.5

= 5.5

Curved surface area of cylindrical pillar is 5.5 m.

Find the cost:

Cost of whitewashing 1m² is Rs 12.50 (Given)

Cost of whitewashing 5.5 m² area = Rs. 12.50 x 5.5 = Rs. 68.75

Thus the cost of whitewashing the pillar is Rs 68.75.

Question 4: It is required to make a closed cylindrical tank of height 1 m and the base diameter of 140 cm from a metal sheet. How many square meters of the sheet are required for the same?

Solution:

Height of cylindrical tank (h) = 1 m

Base radius of cylindrical tank (r) = diameter/2 = 140/2 cm = 70 cm = 0.7 m

Now,

Area of sheet required = Total surface area of tank (TSA) = 2πr(h + r)

=2 x 3.14 x 0.7(1 + 0.7)

= 7.48

Therefore, 7.48 m² metal sheet is required to make required closed cylindrical tank.

Question 5: A solid cylinder has a total surface area of 462 cm². Its curved surface area is one-third of its total surface area. Find the radius and height of the cylinder.

Solution:

Total surface area of a cylinder = 462 cm² (Given)

As per given statement:

Curved or lateral surface area = 1/3 (Total surface area)

⇒ 2πrh = 1/3(462)

⇒ 2πrh = 154

⇒ h = 49/2r ….(1)

Again,

Total surface area = 462 cm²

2πr(h + r) = 462

2πr(49/2r + r) = 462

or 49 + 2r² = 147

or 2r² = 98

or r = 7

Substitute the value of r in equation (1), and find the value of h.

h = 49/2(7) = 49/14 = 7/2

Height (h) = 7/2 cm

Answer: Radius = 7 cm and height = 7/2 cm of the cylinder

Question 6: The total surface area of a hollow cylinder which is open on both the sides is 4620 sq.cm and the area of the base ring is 115.5 sq.cm and height is 7 cm. Find the thickness of the cylinder.

Solution:

Given:

Total surface area of hollow cylinder = 4620 cm²

Height of cylinder (h) = 7 cm

Area of base ring = 115.5 cm²

To find: Thickness of the cylinder

Let ‘r₁’ and ‘r₂’ are the inner and outer radii of the hollow cylinder respectively.

Then, πr₂² – πr₁² = 115.5 …….(1)

And,

2πr₁h +2πr₂h+ 2(πr₂² – πr₁²) = 4620

Or 2πh (r₁ + r₂ ) + 2 x 115.5 = 4620

(Using equation (1) and h = 7 cm)

or 2π7 (r₁ + r₂ ) = 4389

or π (r₁ + r₂ ) = 313.5 ….(2)

Again, from equation (1),

πr₂² – πr₁² = 115.5

or π(r₂ + r₁) (r₂ – r₁) = 115.5

Using result of equation (2),

313.5 (r₂ – r₁) = 115.5

or r₂ – r₁ = 7/19 = 0.3684

Therefore, thickness of the cylinder is 7/19 cm or 0.3684 cm.

Question 7: Find the ratio between the total surface area of a cylinder to its curved surface area, given that height and radius of the tank are 7.5 m and 3.5 m.

Solution:

Height of cylinder (h) = 7.5 m

Radius of cylinder (r) = 3.5 m

We know, Total Surface Area of cylinder (T.S.A) = 2πr(r+h)

And, Curved surface area of a cylinder(C.S.A) = 2πrh

Now, Ratio between the total surface area of a cylinder to its curved surface area is

T.S.A/C.S.A = 2πr(r+h)/2πrh

= (r + h)/h

= (3.5 + 7.5)/7.5

= 11/7.5

= 22/15 or 22:15

Therefore the required ratio is 22:15.

Exercise 19.2 Page No: 19.20

Question 1: A soft drink is available in two packs- (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 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:

(i) Dimensions of a cubical tin can:

Length (L) = 5 cm

Breadth (B) = 4 cm

Height (H) = 15 cm

Capacity of the tin can = Volume of Tin Can = l x b x h cubic units = (5 x 4 x 15) cm³ = 300 cm³

(ii) Radius of the circular end of the plastic cylinder (R) = diametr/2 = 7/2 cm = 3.5 cm

Height of plastic cylinder (H) = 10 cm

Capacity of plastic cylinder = Volume of cylindrical container = πR²H = 22/7 × (3.5)² × 10 cm³ = 385 cm³

From (i) and (ii) results, the plastic cylinder has greater capacity.

Difference in capacity = (385 – 300) cm³ = 85 cm³

Question 2: The pillars of a temple are cylindrically shaped. If 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:

In this case, we have to find the volume of the cylinders.

Given:

Radius of the base of a cylinder = 20 cm

Height of cylinder = 10 m = 1000 cm

Volume of the cylindrical pillar = πR²H

= (22/7×20²×1000) cm³

= 8800000/7 cm³ or 8.87 m³

Therefore, volume of 14 pillars = 14 x 8.87 m³ = 17.6 m³

Question 3: 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 gm.

Solution:

Let r and R be the inner and outer radii of cylindrical pipe.

Inner radius of a cylindrical pipe (r) = 24/2 = 12 cm

Outer radius of a cylindrical pipe (R) = 24/2 = 14 cm

Height of pipe (h) = length of pipe = 35 cm

Mass of pipe = volume x density = π(R² – r²)h

= 22/7(14² – 12²)35

= 5720

Mass of pipe is 5720 cm³

Mass of 1 cm³ wood = 0.6 gm (Given)

Therefore, mass of 5720 cm³ wood = 5720 x 0.6 = 3432 gm = 3.432 kg

Question 4: If the lateral surface of a cylinder is 94.2 cm² and its height is 5 cm, find:

i) radius of its base (ii) volume of the cylinder

[Use π = 3.141]

Solution:

Lateral surface of the cylinder = 94.2 cm²

Height of the cylinder = 5 cm

Let ‘r’ be the radius.

(i) Lateral surface of the cylinder = 94.2 cm²

2 πrh = 94.2

or 2 x 3.14 x r x 5 = 94.2

or r = 3 cm

(ii) Volume of the cylinder = πr²h

= (3.14 x ³² x 5) cm³

= 141.3 cm³

Question 5: The capacity of a closed cylindrical vessel of height 1 m is 15.4 liters. How many square meters of the metal sheet would be needed to make it?

Solution:

Given, The capacity of a closed cylindrical vessel of height 1 m is 15.4 liters.

Height of the cylindrical vessel = 15.4 litres = 0.0154 m³

Let ‘r’ be the radius of the circular ends of the cylinders, then

πr²h = 0.0154 m³

3.14 x r² x 1 = 0.0154 m³

or r = 0.07 m

Again,

Total surface area of a vessel = 2πr(r+h)

= 2(3.14(0.07)(0.07+1)) m²

= 0.470 m²

Question 6: 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:

Radius of cylindrical bowl (R) = diameter/2 = 7/2 cm = 3.5 cm

Height = 4 cm

Now,

Volume of soup in 1 bowl = πr²h

= 22/7×3.5²×4 cm³

= 154 cm³

Volume of soup in 250 bowls = (250 x 154) cm³

= 38500 cm³

= 38.5 liters

Thus, hospital has to prepare 38.5 liters of soup daily in order to serve 250 patients.

Question 7: A hollow garden roller, 63 cm wide with a girth of 440 cm, is made of 4 cm thick iron. Find the volume of the iron.

Solution:

The outer circumference of the roller = 440 cm

Thickness of the roller = 4 cm and

Its height (h) = 63 cm

Let ‘R’ be the external radius and ‘r’ be the inner radius of the roller.

Circumference of roller = 2πR = 440

Or 2πR = 440

2×22/7 x R = 440

or R = 70

And, inner radius ‘r’ is given as

⇒ r = R – 4

⇒ r = 70 – 4

⇒ r = 66

Inner radius is 66 cm

Now, volume of the iron is given as

V = π(R²−r²)h

V = 22/7 (70²−66²)63

V = 107712

Therefore, required volume is 107712 cm³.

Question 8: A solid cylinder has a total surface area of 231 cm². Its curved surface area is 2/3 of the total surface area. Find the volume of the cylinder.

Solution:

Total surface area = 231 cm²

As per given statement: Curved surface area = 2/3(Total surface area)

Curved surface area = 2/3 x 231 = 154

So, Curved surface area = 154 cm² …(1)

We know, Curved surface area of cylinder = 2πrh + 2πr²

Or 2πrh + 2πr² = 231 …..(2)

Here 2πrh is the curved surface area, so using (1), we have

⇒ 154 + 2πr² = 231

⇒ 2πr² = 231- 154

⇒ 2 x 22/7 x r² = 77

⇒ r² = 49/4

or r = 7/2

Find the value of h:

CSA = 154 cm²

⇒ 2πrh = 154

⇒ 2 x 22/7 x 7/2 x h = 154

⇒ h = 154/22

⇒ h = 7

Now,

Find Volume of the cylinder:

V = πr²h

= 22/7 x 7/2 x 7/2 x 7

= 269.5

The volume of the cylinder is 269.5 cm³

Question 9: The cost of painting the total outside surface of a closed cylindrical oil tank at 50 paise per square decimetre is Rs 198. The height of the tank is 6 times the radius of the base of the tank. Find the volume corrected to 2 decimal places.

Solution:

Let ‘r’ be the radius of the tank.

As per given statement: Height (h) = 6(Radius) = 6r dm

Cost of painting for 50 paisa or Rs 1/2 per dm² = Rs 198 (Given)

⇒ 2πr(r+h) × 1/2 = 198

⇒ 2×22/7×r(r+6r) × 1/2 = 198

⇒ r = 3 dm

And, h = (6 x 3) dm = 18 dm

Now,

Volume of the tank = πr²h = 22/7×9×18 = 509.14 dm³

Question 10: The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes and the ratio of their curved surfaces.

Solution:

Let the radius of the cylinders be 2x and 3x and the height of the cylinders be 5y and 3y.

Question 11: 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:

Total surface area (T.S.A) = 616 cm² (given)

Let r be the radius of cylinder and h be the radius of cylinder.

As per given statement:

(curved surface area / (total surface area) = 1/2

or CSA = 12 TSA

CSA = 12 x 616 = 308

⇒ CSA = 308 cm²

Now,

TSA = 2πrh + 2πr²

⇒ 616 = CSA + 2πr²

⇒ 616 = 308 + 2πr²

⇒ 2πr² = 616 – 308

⇒ 2πr² = 308/2π

⇒ r² = 49

or r = 7 cm …(1)

As, CSA = 308 cm²

2πrh = 308

⇒ 2 x 22/7 x 7 x h = 308

(using (1))

⇒ h = 7 cm

Now,

Volume of cylinder = πr²h

= 22/7 x 7 x 7 x 7

= 1078

Therefore, Volume of cylinder is 1078 cm².

Question 12: The curved surface area of a cylinder is 1320 cm² and its base had diameter 21 cm. Find the height and volume of the cylinder.

Solution:

Curved surface area of a cylinder = 1320 cm²

Let, r be the radius of the cylinder and h be the height of the cylinder.

⇒ r = diameter/2 = 21/2 cm = 10.5 cm

We know, Curved surface area(CSA) = 2πrh

So, 2πrh = 1320

⇒ 2x 22/7 x 10.5 x h = 1320

or h = 20 cm

Now,

Volume of cylinder = πr²h

= 22/7 x 10.5 x 10.5 x 20

= 6930

Thus, Volume of cylinder is 6930 cm².

Question 13: The ratio between the radius of the base and the height of a cylinder is 2:3. Find the total surface area of the cylinder, if its volume is 1617cm³.

Solution:

Let, r be the radius of the cylinder and h be the height of the cylinder.

As per statement: r:h = 2:3

Then, radius = 2x cm and height = 3x cm

Volume of cylinder = πr²h

And Volume of cylinder= 1617 cm^3 (given)

So, 1617= 22/7 (2x)² 3x

1617 = 22/7 (12 x³ )

x³ = 343/8

or x = 7/2

or x = 3.5 cm

Now, radius, r = 2 x 3.5 = 7 cm and

Height = 3x = 3 x 3.5 = 10.5 cm

Now,

Total surface area of cylinder = 2πr(h+r)

= 2 x 22/7 x 7(10.5+7)

= 770

Thus, Total surface area of cylinder is 770 cm².

Question 14: A rectangular sheet of paper, 44 cm x 20 cm, is rolled along its length of form cylinder. Find the volume of the cylinder so formed.

Solution:

Length of a rectangular sheet = 44 cm

Height of a rectangular sheet = 20 cm

Now, 2πr = 44

r = 44/2π

r = 44 x 1/2 x 7/22

or r = 7 cm

Now,

Volume of cylinder = π r²h

= 22/7 x 7 x 7 x 20

= 3080

So, Volume of cylinder is 3080 cm³.

Question 15: The curved surface area of cylindrical pillar is 264 m² and its volume is 924 m³. Find the diameter and the height of the pillar.

Solution:

Let, r be the radius of the cylindrical pillar and h be the height of the cylindrical pillar

Curved surface area of cylindrical pillar = CSA = 264 m² (Given)

So, 2πrh = 264

or πrh = 132 …(1)

Again,

Volume of the cylinder = 924 m³ (given)

πr² h= 924

or πrh(r) = 924

Using equation (1)

132 r = 924

or r = 924/132

or r = 7m

Substitute value of r value in equation (1)

22/7 x 7 x h = 132

Or h = 6m

Therefore, diameter = 2r = 2(7) = 14 m and height = 6 m

Exercise VSAQs Page No: 19.27

Question 1: Write the number of surfaces of a right circular cylinder.

Solution:

There are 3 surfaces in a cylinder.

Question 2: Write the ratio of total surface area to the curved surface area of a cylinder of radius r and height h.

Solution:

Ratio of total surface area to the curved surface area of a cylinder of radius r and height h can be written as:

RD Sharma Solutions for Chapter 19 Surface Area and Volume of A Right Circular Cylinder

In this 19th chapter of Class 9 RD Sharma Solutions students will study important concepts listed below:

• Right Circular Cylinder Introduction
• Some important terms definition – Base, Axis, Radius, Height and Lateral Surface.
• Surface Area of a Cylinder
• Volume of a Cylinder