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.

The radii of two cylinders are in the ratio $2 : 3$ and their heights are in the ratio $5:3$.

To do:

We have to find the ratio of their volumes and the ratio of their curved surfaces.

Solution:

Ratio in radii of two cylinders $= 2:3$

Ratio in their heights $= 5:3$

Let the radius of the first cylinder $(r_1) = 2x$ and the radius of the second cylinder $(r_2) = 3x$

Height of the first cylinder $(h_1) = 5y$ and the height of the second cylinder $(h_2) = 3y$

Volume of the first cylinder $= \pi r^2h$

$= \pi (2x)^2 \times 5y$

$= 20\pi x^2y$

Volume of the second cylinder $= \pi (3x)^2 \times 3y$

$= 27\pi x^2y$

Ratio in their volumes $= 20\pi x^2y : 27\pi x^2y$

$= 20 : 27$

Curved surface area of the first cylinder $= 2\pi rh$

$= 2\pi \times 2x \times 5y$

$=20\pi xy$

Curved surface area of the second cylinder = 2\pi \times 3x \times 3y$

$= 18\pi xy$

Therefore,

Ratio in their curved surface areas $= 20\pi xy : 18\pi xy$

$= 10 : 9$