Question

If the height of a cylinder is doubled, by what number must the radius of the base be multiplied so that the resulting cylinder has the same volume as the original cylinder?

Options

• 4

• \[\frac{1}{\sqrt{2}}\]

   

• 2

• \[\frac{1}{2}\]

   

Answer 1

Let V₁ be the volume of the cylinder with radius r₁ and height h₁, then

 `V_1 = pir_1^2 h_1`……. (1)

Now, let V₂ be the volume after changing the dimension, then

` r_2 = xr_1 , h_2 = 2h_1`

So,

`V_2 = pi r_2^2 h_2 = pi xx (xr_1)^2 xx 2h_1`

`⇒ V_2 = 2 xx pi  x^2  r_1^2  h_1`

It is given that V₁ =V₂Therefpre,

`V_1 = V_2`

`⇒ pi r_^2 h_1 = 2 pi x^2  r_1^2 h_1`

`⇒ x^2  = 1/2 r_1^2`

`⇒ x = 1/sqrt(2) r_1`