The radii of the bases of two right circular cones of same height are r₁ and r₂ respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms of r₁, r₂ and R.

Let height of each cone be h cm.

Given,

Cones are melted and recasted into a sphere.

∴ Volume of cone with radii r₁ + Volume of cone with radii r₂ = Volume of sphere

$\Rightarrow \dfrac{1}{3}πr$1^2h + \dfrac{1}{3}πr2^2h = \dfrac{4}{3}πR^3 \\[1em] \Rightarrow \dfrac{πh}{3}(r1^2 + r2^2) = \dfrac{π}{3} \times 4R^3 \\[1em] \Rightarrow h(r1^2 + r2^2) = 4R^3 \\[1em] \Rightarrow h = \dfrac{4R^3}{r1^2 + r2^2}.

Hence, h = $\dfrac{4R^3}{r$1^2 + r2^2}.