# Class 9 RD Sharma Solutions – Chapter 19 Surface Area And Volume of a Right Circular Cylinder – Exercise 19.2 | Set 1

• Last Updated : 07 Apr, 2021

### (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 :

Given that,

The tin can with a rectangular base:

Length= 5 cm,

Height = 15 cm

The plastic cylinder with circular base:

Diameter = 7cm

So the radius of the base = 7/2 cm = 3.5 cm

Height = 10 cm

Now we find the volume of both cans:

Capacity of the tin can = l × b × h = (5 × 4 x 15) cm3

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

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

Hence, the plastic cylinder has greater capacity.

### 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:

Given that,

Radius of the base of a cylindrical pillar= 20 cm

The height of the cylindrical pillar = 10 m

Find how much concrete mixture would be required to build 14 such pillars

So

Volume of the cylindrical pillar = πR2H

= (22/7 × 202 × 1000)

= 8800000/7

= 8.8/7 m3

The volume of 14 pillars = 8.8/7 × 14 = 17.6 m3

Hence, the volume of the 14 pillars = 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:

Given that,

The inner diameter of a cylindrical wooden pipe(d1) = 24 cm

So, the inner radius of a cylindrical pipe(r1) = 24/2 = 12 cm

The outer diameter of a cylindrical wooden pipe(d2) = 28 cm

So, the outer radius of a cylindrical pipe(r2) = 28/2 = 14 cm

Height of cylindrical pipe (h) = 35 cm

Find the mass of the pipe, if 1 cm3 of wood has a mass of 0.6 gm

So,

The Mass of pipe = Volume x density

= π(r22 – r12)

= 22/7 x (142 − 122) x 35 = 5720 cm3

Mass of 1 cm3 wood = 0.6 gm

Mass of 5720 cm3 wood = 5720 × 0.6 = 3432 gm = 3.432 kg

### (ii) volume of the cylinder [Use pi = 3.141]

Solution :

Given that,

The lateral surface of a cylinder = 94.2 cm2

The hight of the cylinder = 5cm

(i) Find the radius of its base

Let’s assume that the radius of cylinder be ‘r’

Curved surface of the cylinder = 2πrh

94.2 = 2 (3.14)r(5)

r = 3 cm

Hence, the radius of the cylinder is 3 cm

(ii) As we know that

The volume of the cylinder = πr2

= (3.14 × 32 × 5)

= 141.3 cm3

Hence, the volume of the cylinder is 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 that,

The height of the cylindrical vessel = 1m

The capacity/volume of the cylinder = 15.4 liters = 0.0154 m3     (As we know 1m3 = 1000 liter)

Let’s assume that the radius of the circular ends of the cylinders be ‘r’

So the volume of the cylinder is

V = πr2

0.0154 = (31.4)r2(1)

r = 0.07 m

Now we find the total surface area of a vessel:

TSA = 2πr(r + h)

= 2(3.14 x (0.07) x (0.07 + 1)) = 0.4703 m2

Hence, we need 0.4703 m2 of the metal sheet

### 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:

Given that,

The diameter of cylindrical bowl = 7 cm = 3.5 cm

So, the radius = 7/2 cm = 3.5 cm

The bowl is filled with soup to a height =4cm

Now we find the volume soup in 1 bowl

V = πr2h

= 22/7 × 3.52 × 4 = 154 cm3

So the volume soup in 250 bowl

V = (250 × 154) = 38500 cm3 = 38.5 liter

Hence, the soup, hospital has to prepare daily to serve 250 patients is 38.5 liter.

### 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:

Given that,

Garden roller height = 63 cm,

Garden roller outer circumference = 440 cm,

Garden roller thickness = 4 cm

Find the volume of iron.

So, let’s assume that the R be the external radius and the inner radius be ‘r’

2πR = 440

2 x 22/7 x R = 440

R = 70

Now we find the value of inner radius:

r = R – 4

70 – 4 = 66cm

Now we find the volume of the iron:

V = π (R2 − r2) x h

= 22/7 x (702 − 662) x 63

= 22/7 x 4 x 136 x 63 = 107712 cm3

Hence, the volume of the iron is 107712 cm3

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

Solution:

Given that,

Total surface area = 231cm2

Curved surface area = 2/3 x (Total Surface Area)

So,

Curved surface area = 2/3 x 231 = 154

As we know that,

the total surface area of cylinder = 2πrh + 2πr2

2πrh + 2πr2 = 231 —————-(i)

Where, 2πrh is the curved surface area, So

154 + 2πr2 = 231

2πr2 = 231 – 154

2πr2 = 77

2 x 22/7 x r2 = 77

r2 = (7×7) / (2×2)

r = 7/2

The radius of cylinder = 7/2

Now we find the height of the cylinder

So, as we know that

Curved surface area = 2πrh

2πrh = 154

2 x 22/7 x 7/2 x h = 154

h = 154/22 = 7

So, the height of cylinder = 7

Now we find the volume of the cylinder:

Volume = πr2h

= 22/7 x 7/2 x 7/2 x 7 = 269.5 cm3

So, 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’s assume that the radius of the tank = r dm

So, the height of the tank(h) = 6r dm

It is given that the cost of painting = 50 paisa per dm

So, the total cost of painting = Rs 198

= 2πr(r + h) = 198

= 2 × 22/7 × r(r + 6r) × 1/2 = 198

r = 3 dm

Hence the radius of the tank is 3 dm

Therefore, h = (6 × 3) dm = 18 dm

As we know that,

Volume of the tank = πr2

= 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:

Given that the ratio of the radii of two cylinders = 2:3

The ratio of the heights two cylinders = 5:3

So, let’s assume that the radius of the two cylinders are 2x and 3x

The height of the two cylinders is 5y and 3y

Find: The ratio of their volumes and the ratio of their curved surfaces

So, for the ratio of their volumes:

We have

Volume of cylinder A/ Volume of cylinder B = π (r)2 h/π (R)2

= π (2x)2 5y/π (3x)2 3y = 20/27

Hence, the ratio of the volumes of two cylinders are 20:27.

So, for the ratio of their surface area:

We have

Surface area of cylinder A / Surface area of cylinder B = 2πrh/2πRH

= (2π × 2x × 5y) / (2π × 3x × 3y) = 10 / 9

Hence, the ratio of the surface area of two cylinders are 10:9.

### 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:

Given that

Total surface area (TSA) = 616 cm2

The ratio between the curved surface area and the total surface area of a right circular cylinder = 1 : 2

Find: the volume of the cylinder

According to the question

Curved Surface Area / Total Surface Area = 1/2

CSA = 1/2 x TSA

CSA = 1/2 x 616

CSA = 308 cm2

Now, we find the total surface area

TSA = 2πrh + 2πr2

616 = CSA + 2πr2

616 = 308 + 2πr2

2πr2 = 616 – 308

2πr2 = 308

πr2 = 308/2

r2 = 308/2π

r = 7 cm

Since, CSA = 308 cm2

2πrh = 308

2 x 22/7 x 7 x h = 308

h = 7cm

Now we find the volume of cylinder

V = πr2 x h

= 22/7 x 7 x 7 x 7

= 22 x 49

= 1078 cm3

Hence, the 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:

Given that

The curved surface area of a cylinder = 1320 cm

Diameter of its base = 21 cm

So, radius = 21/2 = 10.5 cm

r = 21/2 = 10.5 cm

Find: the height and volume of the cylinder.

So, the curved surface area of a cylinder is

CSA = 2πrh

2 x 22/7 x 10.5 x h = 1320

h = 1320/66 = 20 cm

So the height of the cylinder is 20 cm

Now we find the volume of cylinder

V = πr2 h

= 22/7 x 10.5 x 10.5 x 20

= 22 x 1.5 x 10.5 x 20 = 6930 cm3

Hence, the 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 1617 cm3.

Solution:

Given that,

The volume of the cylinder = 1617 cm3

The ratio between the radius of the base and the height of a cylinder = 2:3

r/h = 2/3

r = 2/3 x h  ——————–(i)

Find: The total surface area of the cylinder

So, we find the volume of cylinder

V = πr2 h

1617 = 22/7 x (2/3 x h)2 x h

1617 = 22/7 x (2/3 x h)3

h3 = (1617 x 7 x 3) / 22 x 4

h = 10.5 cm

From, eqn. (i), we get

r = 2/3 x 10.5 = 7 cm

Now we find the total surface area of cylinder

TSA = 2πr (h + r)

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

= 44 x 17.5

= 770 cm2

Hence, the 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:

Given that,

The dimensions of the rectangular sheet of paper = 44 cm x 20 cm

So,

Length = 44 cm,

Height = 20 cm

Find: The volume of the cylinder

Curved Surface Area = 2πr

2πr = 44

r = 44/2π

r = 44/2π = 7 cm

Hence, the radius of the cylinder is 7 cm

Now, we find the volume of cylinder

V = πr2 h

= 22/7 x 7 x 7 x 20

= 154 x 20 = 3080 cm3

Volume of cylinder is 3080 cm3

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

Solution:

Given that,

The curved surface area of the cylindrical pillar = 264 m2

The volume of the cylindrical pillar = 924 m3

We have to find the diameter and the height of the pillar

So,

Volume of the cylinder

V = πr2h

π x r2 x h = 924

πrh(r) = 924

πrh = 924/r

As we know that the curved surface area of the cylinder

CSA = 2πrh

264 = 2πrh …(1)

Substitute πrh in this eq and we get,

2 x 924/r = 264

r = 1848/264 = 7 m

Substitute r value in eq (i) and we get,

2 x 22/7 x 7 x h = 264

h = 264/44 = 6 m

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

### Question 16. Two circular cylinders of equal volumes have their heights in the ratio 1 : 2. Find the ratio of two radii.

Solution:

Let’s assume that we have two cylinders,

So, the radius of the cylinders = r1, r2

The height of the cylinders = h1, h2

The volume of the cylinders = v1, v2

According to the question

It is given that the h1/h2 = 1/2 and v1 = v2

We have to find the ratio of two radii

So,

v1/v2 = (r1/r2)2 x (h1/h2)

As v1 = v2

v1/v1 = (r1/r2)2 x (1/2)

1 = (r1/r2)2 x (1/2)

(r1/r2)2 = (2/1)

(r1/r2) = √2 / 1

Hence, the ratio of the radii are √2:1

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