# The radii of the bases of two right circular solid cones of the same height are r1 & r2. The cones are melted & recasted into a solid sphere if radius R . show that the height of each cone is given by h=4R^3÷r1^2+r2^2 ?

Feb 27, 2018

See below. Quite simple really.

#### Explanation:

Volume of cone 1; $\pi \cdot {r}_{1}^{2} \cdot \frac{h}{3}$
Volume of cone 2: $\pi \cdot {r}_{2}^{2} \cdot \frac{h}{3}$
Volume of the sphere:$\frac{4}{3} \cdot \pi \cdot {r}^{3}$

So you have:

$\text{Vol of sphere" = "Vol of cone 1" + "Vol of cone 2}$

$\frac{4}{3} \cdot \pi \cdot {R}^{3} = \left(\pi \cdot {r}_{1}^{2} \cdot \frac{h}{3}\right) + \left(\pi \cdot {r}_{2}^{2} \cdot \frac{h}{3}\right)$

Simplify:

$4 \cdot \pi \cdot {R}^{3} = \left(\pi \cdot {r}_{1}^{2} \cdot h\right) + \left(\pi \cdot {r}_{2}^{2} \cdot h\right)$

$4 \cdot {R}^{3} = \left({r}_{1}^{2} \cdot h\right) + \left({r}_{2}^{2} \cdot h\right)$

$h = \frac{4 {R}^{3}}{{r}_{1}^{2} + {r}_{2}^{2}}$