An ellipsoid has radii with lengths of 8 , 7 , and 7 . A portion the size of a hemisphere with a radius of 5  is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Nov 20, 2016

The remaining volume is $= 1380.2$

Explanation:

The volume of an ellipsoid is ${V}_{e} = \left(\frac{4}{3}\right) \pi a b c$

The volume of a hemisphere is ${V}_{h} = \left(\frac{2}{3}\right) \pi {r}^{3}$

The remaining volume ${V}_{r} = {V}_{e} - {V}_{h}$

${V}_{r} = \left(\frac{2 \pi}{3}\right) \left(2 a b c - {r}^{3}\right)$

${V}_{r} = \left(\frac{2 \pi}{3}\right) \left(2 \cdot 8 \cdot 7 \cdot 7 - {5}^{3}\right)$

$= \frac{1318 \pi}{3} = 1380.2$

Nov 20, 2016

$\frac{1318 \pi}{3}$

Explanation:

Volume of an ellipsoid equals $\frac{4}{3} \pi a b c$. In this case it would be $\frac{4}{3} \pi \cdot 8 \cdot 7 \cdot 7 = \frac{1568 \pi}{3}$

Volume of an hemisphere is $\frac{2}{3} \pi {r}^{3}$. In this case it would be $\frac{2}{3} \pi {5}^{3} = \frac{250 \pi}{3}$

The volume of the remaining solid would be $\frac{1568 \pi}{3} - \frac{250 \pi}{3} = \frac{1318 \pi}{3}$