Question

Two cones have their heights in the ratio 1 : 3 and radii 3 : 1. What is the ratio of their volumes?

Answer 1

Let the radius of the cone is 3x and x,

And the height of the cone is y and 3y.

Then,

Volume of the first cone

`v_1 = 1/3 πr^2h`

= `1/3 π(3x)^2y`

= `1/3 π9x^2y`

= 3πx²y   ...(1)

Volume of the second cone

`v_2 = 1/3 π(x)^2 xx 3y`

= πx²y   ...(2)

Then the radius of their volume

`v_1/v_2 = (3πx^2y)/(πx^2y)`

Or

`v_1/v_2 = 3 : 1`

v₁ : v₂ = 3 : 1

Answer 2

Let h₁ and h₂ be height and r₁, r₂ be radii of two cones, then ratio of their volumes

= `(1/3 πr_1^2h_1)/(1/3 πr_2^2h_2)`

Given: `h_1/h_2 = 1/3` and `r_1/r_2 = 3/1`

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

= `(3/1)^2 (1/3)`

= `3/1`

Hence, ratio of their volumes is 3 : 1.