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

Data given is as follows:

Ratio of radii of two cylinders = 2:3

Ratio of heights of two cylinders = 5:3

We have to find out the following:

(i) Ratio of the volumes of the two cylinders

(ii) Ratio of the Curved Surface Area of the two cylinders

Let `r_1`and `r_2`be the radii of the two cylinders respectively.

Let `h_1` and `h_2`be the heights of the two cylinders respectively.

Therefore we have,

`r_1/r_2 = 2/3`

`h_1/h_2 = 5/3`

(i) Since we have to find the ratio of the volumes of the two cylinders, we have

`("Volume of cylinder 1")/("Volume of cylinder 2") = (pi r_1^2h_1)/(pir_2^2h_2)`

`=(r_1/r_2)^2 (h_1/h_2)`

`=(2/3)^2 (5/3)`

`("Volume of cylinder 1")/("Volume of cylinder 2") = (20/27)`

(ii) Since we have to find the ratio of the curved surface areas of the two cylinders, we have,

`("Curved Surface Area of cylinder 1")/("Curved Surface Area if cylinder 2") = (2pir_1h_1)/(2pir_2h_2)`

`=(r_1/r_2) (h_1/h_2)`

`=(2/3)(5/3)`

`("Curved Surface Area of cylinder 1")/("Curved Surface Area if cylinder 2") = (10/9)`