Question

The radii of two right circular cylinders are in the ratio 1:3 and their heights are in the ratio 4:7. Calculate the ratio of their curved surface areas and also the ratio of their volumes.​

Answer 1

Explanation:

Let the radii of 2 cylinders be 2r and 3r respectively and their heights be 5h and 4h respectively.

Let S

1

and S

2

be the curved surface areas of two cylinders.

S

1

→ CSA of cylinder of radius 2r and height 5h=S

1

=2×π×2r×5h=20πrh

S

2

→ CSA of cylinder with radius 3r and height 4h=S

2

=2×π×3r×4h=24πrh

∴

S

2

S

1

=

24πrh

20πrh

=

6

5

∴S

1

:S

2

=5:6.

Answer 2

Given data : The radii of two right circular cylinders are in the ratio 1:3 and their heights are in the ratio 4:7.

To find : Calculate the ratio of their curved surface areas and also the ratio of their volumes.

Solution : Let the radii of two circular cylinders be 1r and 3r and thier heights be 4h and 7h respectively.

Here, for first cylinder

➜ Radius, r = 1r and Height, h = 3h

For second cylinder

➜ Radius, r = 4r and Height, h = 7h

Now,

➜ Curved surface area of the first cylinder

= 2πrh

➜ Curved surface area of the first cylinder

= 2π * 1r * 3h

Here, CSA of 1st cylinder = 2π * 1r * 3h

and

➜ Curved surface area of the second cylinder

= 2πrh

➜ Curved surface area of the second cylinder

= 2π * 4r * 7h

Here, CSA of 2cd cylinder = 2π * 4r * 7h

Now,

➜ CSA of 1st cylinder : CSA of 2cd cylinder

= 2π * 1r * 3h/2π * 4r * 7h

➜ CSA of 1st cylinder : CSA of 2cd cylinder

= 1r * 3h/4r * 7h

➜ CSA of 1st cylinder : CSA of 2cd cylinder

= 4rh/28rh

➜ CSA of 1st cylinder : CSA of 2cd cylinder

= 1rh/7rh

➜ CSA of 1st cylinder : CSA of 2cd cylinder

= 1/7

➜ CSA of 1st cylinder : CSA of 2cd cylinder

= 1:7

Now,

➜ Volume of the first cylinder = πr²h

➜ Volume of the first cylinder = π * (1r)² * 3h

➜ Volume of the first cylinder = π * r² * 3h

Here, V of 1st cylinder = π * r² * 3h

and

➜ Volume of the second cylinder = πr²h

➜ Volume of the second cylinder = π * (4r)² * 7h

➜ Volume of the first cylinder = π * 16r² * 7h

Here, V of 2cd cylinder = π * 16r² * 7h

Now,

➜ V of 1st cylinder : V of 2cd cylinder

= π * r² * 3h/π * 16r² * 7h

➜ V of 1st cylinder : V of 2cd cylinder

= r² * 3h/ 16r² * 7h

➜ V of 1st cylinder : V of 2cd cylinder

= 3r²h/112r²h

➜ V of 1st cylinder : V of 2cd cylinder

= 3/112

➜ V of 1st cylinder : V of 2cd cylinder

= 3 : 112

Answer : Hence, the the ratio of the curved surface areas and the ratio of the volumes of the circular cylinders are 1:7 and 3:112 respectively.