Question

The radii of the bases of two solid right circular cones of same height are $r_1$ and $r_2$ respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms of $r_1,r_2$ and R.

Answer 1

Given radii of the bases of two right circular solid cones of same height r₁ and r₂ respectively.

Volume of first cone =$\frac{1}{3}\times \pi (r_1{)}^2\times h$

Volume of second cone = $\frac{1}{3}\times \pi (r_2{)}^2\times h$

According to the question, the 2 cones are melted and recast to form a sphere of radius R.

Therefore, volume of sphere = volume of first cone + volume of second cone .

$\frac{4}{3}\pi R^3$= ($\frac{1}{3}\times \pi (r_1{)}^2\times h$ )+($\frac{1}{3}\times \pi (r_2{)}^2\times h$)
$4R^3$= ($(r_1{)}^2\times h$ )+($(r_2{)}^2\times h$)
$h=\frac{4R^3}{r_1^2+r_2^2}$