Question

The ratio of the radii of two right circular cones of the same height is 1 : 3. Find the ratio of their curved surface area when the height cone is 3 times the radius of the smaller cone.

Answer 1

Let the radius of the first cone be ‘x’ and the Height of the cone be 3x

l = `sqrt("h"^2 + "r"^2)`

= `sqrt((3x)^2 + x^2)`

= `sqrt(10x^2)`

C.S.A. of the first cone = πrl sq.units

= `pi xx x sqrt(10x^2)`

= `pi x^2 sqrt(10)`

The radius and the height of the second cone is 3x  ...(Given)

l = `sqrt((3x)^2 + (3x)^2)`

= `sqrt(9x^2 + 9x^2)`

= `sqrt(18x^2)`

C.S.A of the second one

= `pi xx 3x xx sqrt(18x^2)`

= `pi xx 3x^2 xx sqrt(9 xx 2)`

= `pi  9x^2 sqrt(2)`

Ratio of the curves surface area

= `pix^2 sqrt(10) : 9 pi x^2 sqrt(2)`

= `sqrt(10) : 9sqrt(2)`

= `sqrt(2) xx sqrt(5) : 9sqrt(2)`

= `sqrt(5) : 9`