Class 8 RD Sharma Solutions- Chapter 22 Mensuration III (Surface Area And Volume Of Right Circular Cylinder) – Exercise 22.2 | Set 1

Question 1.  Find the volume of a cylinder whose

(i) r = 3.5 cm, h = 40 cm

(ii) r = 2.8 m, h = 15 m

Solution: 

(i) Given that,

         r = 3.5 cm,

         h = 40 cm

         As we know that Volume of a cylinder = πr²h

         = 22/7 × 3.5 × 3.5 × 40 = 1540 cm³

(ii) Given that,

         r = 2.8 m,

         h =15 m

         As we know that Volume of a cylinder = πr²h

         = 22/7 × 2.8 × 2.8 × 15 = 369.6 m³

Question 2. Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are :

(i) d = 21 cm, h = 10 cm

(ii) d = 7 m, h = 24 m

Solution:

(i) Given that,

         d = 21cm,

         r = d/2 = 21/2cm,

         h = 10 cm.

         As we know that Volume of a cylinder = πr²h

         = 22/7 × 21/2 × 21/2 × 10 = 3465 cm³

(ii) Given that,

         d = 7 m²

         r = d/2 = 7/2m

         h = 24 m

         As we know that Volume of a cylinder = πr²h

         = 22/7 × 7/2 × 7/2 × 24 = 924 m³

Question 3. The area of the base of a right circular cylinder is 616 cm² and its height is 25 cm. Find the volume of the cylinder.

Solution: 

Given that,

Area of base of right circular cylinder = 616 cm²

Height of cylinder = 25 cm

Let us assume that radius of cylinder is ‘r’ cm

As we know that Area of base of right circular cylinder = πr²,

πr² = 616

22/7 × r² = 616

r² = 616 × 7/22 = 196

r = √196 = 14cm

As we know that, Volume of cylinder = Area of base of right circular cylinder × height

= 616 × 25 = 15400 cm³

Question 4. The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find the volume of the cylinder.

Solution:

Given that,

Circumference of base of cylinder = 88 cm

Height of cylinder = 15 cm

Let us assume that ‘r’ be the radius of the cylinder.

As we know that Circumference of base of cylinder = 2πr,

2πr = 88

2 × 22/7 × r = 88

r = 88 × 7 / 2 × 22 = 616/44 = 14cm

Radius of cylinder = 14 cm

We know that Volume of cylinder = πr²h

= 22/7 × 14 × 14 × 15 = 9240 cm³

Question 5. A hollow cylindrical pipe is 21 dm long. Its outer and inner diameters are 10 cm and 6 cm respectively. Find the volume of the copper used in making the pipe.

Solution:

Given that,

Length of cylinder = 21 dm = 210 cm

Outer diameter = 10 cm

Outer radius, R = 10/2 = 5cm

Inner diameter = 6 cm

Inner radius, r = 6/2 = 3cm

As we know that Volume of copper used in making the pipe(hollow) = π (R² – r²)h²

= 22/7 (52 – 32) 210

= 22/7 (25-9) 210 = 10560 cm³

Question 6. Find the (i) curved surface area (ii) total surface area and (iii) volume of a right circular cylinder whose height is 15 cm and the radius of the base is 7 cm.

Solution:

Given that,

Height of cylinder = 15 cm

Radius of base = 7 cm

(i) We know that formula of Curved surface area = 2πrh

= 2 × 22/7 × 7 × 15 = 660 cm²

(ii) We know that formula of Total surface area = 2πr(h+r)

= 2 × 22/7 × 7 (15+7) = 968 cm²

(iii) We know that formula of Volume of cylinder = πr²h

= 22/7 × 7 × 7 × 15 = 2310 cm³

Question 7. The diameter of the base of a right circular cylinder is 42 cm and its height is 10 cm. Find the volume of the cylinder.

Solution: 

Given that,

Diameter of base of cylinder = 42 cm

Radius of base = d/2 = 42/2 = 21cm

Height = 10 cm

We know that Volume of cylinder = πr²h

= 22/7 × 21 × 21 × 10 = 13860 cm³

Question 8. Find the volume of cylinder, the diameter of whose base is 7 cm and height being 60 cm. Also, find the capacity of the cylinder in litres.

Solution: 

Given that,

Diameter of base = 7 cm

Radius of base = d/2 = 7/2 cm

Height of cylinder = 60 cm

As we know that Volume of cylinder = πr²h

= 22/7 × 7/2 × 7/2 × 60 = 2310 cm³

Capacity of cylinder in liters = 2310 / 1000 = 2.31 liters.

Question 9. A rectangular strip 25 cm× 7 cm is rotated about the longer side. Find the volume of the solid, thus generated.

Solution: 

Given that,

Dimensions of Rectangular Strip = 25 cm × 7 cm

When it rotated about its longer side then it will become,

Radius of base = 7 cm

Height of cylinder = 25 cm

As we know that Volume of cylinder = πr²h

= 22/7 × 7 × 7 × 25 = 3850 cm³

Question 10. A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.

Solution:

Given that,

Dimensions of rectangular sheet = 44cm × 20cm

When it rolled along its length it will become,

Radius of base = length/2π

= 44×7 / 2×22 = 7cm

Height of cylinder = 20 cm

We know that Volume of cylinder = πr²h

= 22/7 × 7 × 7 × 20 = 3080 cm³

Question 11. The volume and the curved surface area of cylinder are 1650 cm³ and 660 cm² respectively. Find the radius and height of the cylinder.

Solution:

Given that,

Volume of cylinder = 1650 cm³

Curved surface area = 660 cm²

Volume of Cylinder/Curved Surface Area = 1650/660

πr²h/ 2πrh = 1650/660

r/ 2 = 5/2

r = 5cm

Surface area = 660 cm²      (Given)

As we know that Surface Area of Cylinder = 2πrh

2πrh = 660

2 × 22/7 × 5 × h = 660

h = 660×7 / 2×22×5

= 4620/220 = 21cm

Hence, Radius is 5cm and height is 21cm.

Question 12. 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.

Solution: 

Given that,

Ratio of radii of two cylinder = 2:3

Radius of 1st Cylinder = r1

Radius of 2nd Cylinder = r2

r1/r2 = 2/3

Ratio of their heights = 5:3

Height of 1st Cylinder = h1

Height of 2nd Cylinder = h2

h1/h2 = 5/3

Volume of 1st Cylinder = v1

Volume of 2nd Cylinder = v2

v1 / v2 = π(r1)²h1 / π(r2)²h2

= 22 × 5 / 32 × 3

= 4×5 / 9×3 = 20/27

Hence, Ratio of Volumes of two Cylinder’s is 20 : 27.

Question 13. 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 of cylinder = 616 cm²

Ratio between Curved Surface Area and Total Surface Area of Cylinder = 1:2

2πrh / 2πr (r+h) = 1/2

h / (r+h) = 1/2

2h =r+h

Hence, r = h

As we know, 2πr (h+r) = 616                

2πr (r+r) = 616

2πr (2r) = 616

4πr² = 616

r² = 616/4π

= 616×7 / 4×22 = 49

r = √49 = 7

Hence, Radius = 7cm and Height = 7cm

As we know that Volume of cylinder = πr²h

= 22/7 × 7 × 7 × 7 = 1078 cm³

Question 14. The curved surface area of a cylinder is 1320 cm² and its base has diameter 21 cm. Find the volume of the cylinder.

Solution: 

Given that,

Diameter of base = 21 cm

Radius of base = d/2 = 21/2 cm

Curved surface area = 1320 cm²

As we know that Curved surface Area of Cylinder = 2πrh

2πrh = 1320

2 × 22/7 × 21/2 × h = 1320

h = 1320×7×2 / 2×22×21

= 18480/924 = 20cm

As we know that Volume of Cylinder = πr²h

= 22/7 × 21/2 × 21/2 × 20 = 6930 cm³

Question 15. 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,

Ratio between radius and height of a cylinder = 2:3

r/h = 2/3

h = 3/2 r

Volume of cylinder = 1617 cm³

As we know that Volume of Cylinder = πr²h

πr²h = 1617

22/7 × r² × 3/2r = 1617

r³ = 1617×7×2 / 22×3 = 343

r = ³√343

= 7cm

Hence, Radius = 7 cm

Height = 3/2r = 3/2 × 7 = 21/2 = 10.5cm

As we know that Total Surface Area of Cylinder = 2πr (r+h)

= 2 × 22/7 × 7 (10.5+7) = 770 cm²

Question 16. The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the diameter and the height of the pillar.

Solution:

Given that,

Curved surface area of cylinder = 264 m²,

Volume = 924 m³,

Volume of Cylinder / Curved Surface Area of Cylinder

πr²h / 2πrh = 924 / 264

r/2 = 924 / 264

r = 924×2 / 264 = 7m

Hence, Radius = 7 m

Diameter of cylinder = 2 × radius = 2×7 = 14m

Curved surface area = 264 m²      (Given)

2πrh = 264

2 × 22/7 × 7 × h = 264

h = 264×7 / 2×22×7 = 6m

Hence, Height of cylinder is 6m & Diameter of cylinder is 14m.

Question 17. Two circular cylinders of equal volumes have their heights in the ratio 1:2. Find the ratio of their radii.

Solution:

Given that,

Ratio of their height = 1:2,

Height of 1st Cylinder = h1,

Height of 2nd Cylinder = h2

h1 / h2 = 1/2

Volume of 1st Cylinder, V1 = Volume of 2nd Cylinder, V2

V1 = V2

π(r1²)h1 = π(r2² )h2

r1² / r2² = 2/1

r1 / r2 = √(2/1) = √2 / 1

Hence, the Ratio of their radii is √2:1

Question 18. The height of a right circular cylinder is 10/5 m. Three times the sum of the areas of its two circular faces is twice the area of the curved surface. Find the volume of the cylinder.

Solution: 

Given that,

Height of cylinder = 10.5 m

3(A+A) = 2 Curved Surface Area (where, A = Circular Area of Box)

3×2A = 2(2πrh)

6A = 4πrh

6πr² = 4πrh

r²/r = 4πh/6π

r = 2/3 h = 2×10.5 / 3 = 7m

As we know that Volume of Cylinder = πr²h

= 22/7 × 7 × 7 × 10.5 = 1617 m³

Question 19. How many cubic meters of earth must be dug-out to sink a well 21 m deep and 6m diameter?

Solution:

Given that,

Height of cylinder = 21m,

Diameter of well = 6m,

Radius of well = d/2 = 6/2 = 3m

We know that Volume of Earth that must be dug out from this well is = πr²h

= 22/7 × 3 × 3 × 21 = 594 m³

Question 20. The trunk of a tree is cylindrical and its circumference is 176 cm. If the length of the trunk is 3 m, find the volume of the timber that can be obtained from the trunk.

Solution: 

Given that,

Length of the trunk = 3m = 300 cm,

Circumference of trunk of tree = 176 cm

As we know that Circumference of Trunk of tree = 2πr

2πr = 176

2 × 22/7 × r = 176

r = 176×7 / 2×22 = 28cm

Hence, Radius = 28cm

We know that Volume of Timber can be obtained from trunk of tree = πr²h

= 22/7 × 28 × 28 × 300 = 7392 cm³ = 0.74 m³

Chapter 22 Mensuration III (Surface Area And Volume Of Right Circular Cylinder) – Exercise 22.2 | Set 2