The curved surface area of a right circular cone is `sqrt(5)` times of its base area . Find the ratio of the height and the length of radius of the cone.

Explanation

Let the radius of the cone is r unit , height is h unit and slant height is l unit .
`therefore` the curved surface area of the cone `= pi r l ` sq- unit `= pi r sqrt(h^2+r^2)` sq-unit.
Also, the area of the base `= pi r^2` sq-unit .
As per question, `pi rsqrt(h^2+r^2)=sqrt(5)pir^2 or , sqrt(h^2+r^2)=sqrt(5)r or , h^2+r^2=5r^2 or , h^2=4r^2or , h^2=(2r)^2, or h=2r, (h)/(r)=(2)/(1)or , h:r =2:1`.
Hence the ratio of the height and radius of the right circular cone is `2:1`.