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
[using π=3.14 ]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)
[1 m = 100 cm]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
[using π = 3.14]= 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
[As radius = half of the diameter] and [1 m = 100 cm]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
[1 m = 100 cm]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)
[Using π = 22/7]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 identity: a^2 – b^2 = (a – b)(a + b)]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
[1m = 100 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
[1m3 = 1000 litres]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 cm3.
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 cm3.
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 Class 9 Maths 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