# An ellipsoid has radii with lengths of 2 , 2 , and 3 . A portion the size of a hemisphere with a radius of 2  is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Jun 12, 2018

$\textcolor{b l u e}{{\text{volume"=(32pi)/3" units}}^{3}}$

#### Explanation:

The volume of an ellipsoid is given by:

$V = \frac{4}{3} \pi \cdot a \cdot b \cdot c$

Where $a , b , c$ are the radii of the ellipsoid.

The volume of a hemisphere is half the volume of a sphere:

Volume of a sphere is:

$V = \frac{4}{3} \pi {r}^{3}$

So volume of hemisphere is:

$V = \frac{2}{3} \pi {r}^{3}$

To find the volume of the ellipsoid when the hemisphere is removed, we just find the volume of the ellipsoid and subtract the volume of the hemisphere:

$V = \frac{4}{3} \pi \cdot a \cdot b \cdot c - \frac{2}{3} \pi {r}^{3}$

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

Substituting values:

$V = \frac{2}{3} \pi \left(2 \cdot 2 \cdot 2 \cdot 3 - {2}^{3}\right)$

$V = \frac{2}{3} \pi \left(16\right)$

$V = \frac{32 \pi}{3}$