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.

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Solution

Given radii of the bases of two right circular solid cones of same height r₁and r₂respectively.

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

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

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 .

$\frac{4}{3} π R^{3}$ = ( $\frac{1}{3} \times π (r_{1})^{2} \times h$ )+( $\frac{1}{3} \times π (r_{2})^{2} \times h$ )
$4 R^{3}$ = ( $(r_{1})^{2} \times h$ )+( $(r_{2})^{2} \times h$ )
$h = \frac{4 R^{3}}{r_{1}^{2} + r_{2}^{2}}$

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