Question

Find ratio of height of cylinder when their lateral surface area is equal n the ratio of the radii is 2:3

Answer 1

Let the radius of first cylinder be x and second be y
Let the height of first cylinder be a and second cylinder be b

Given that Ratio of their radii = 2:3


=>
$\frac{x}{y} = \frac{2}{3} \\ \\ = > x = \frac{2y}{3}$

Now given that the lateral surface area of them is equal.

Lateral surface area = 2πrh

So for first cylinder, lateral surface area =
$2\pi xa$

For second cylinder, lateral surface area =
$2\pi yb$

Now x = 2y/3

and the lateral surface areas are equal

=>
$2\pi \frac{2y}{3} a = 2\pi yb$

cancel 2π and y from both sides as they are common

=>
$\frac{2}{3} a = b$


So ratio of their heights

= a/b

b = 2/3 × a

=> ratio =
$\frac{a}{b} \\ \\ = > a \div \frac{2}{3} a \\ \\ = > a \times \frac{3}{2a} \\ \\ = \frac{3}{2} \\ = > \frac{a}{b} = \frac{3}{2}$


So ratio of their heights = 3:2

Answer :- 3 : 2