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

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 :

Given that,

The tin can with a rectangular base:

Length= 5 cm,

Breadth = 4 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

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

(i) the radius of its base

(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 = πr²h 

= (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 = πr²h 

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 dm² 

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 = 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: 

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 H 

= 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 cm² 

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