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

Jun 6, 2018

$\textcolor{b l u e}{\text{Volume} = \frac{56 \pi}{3}}$

#### 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:

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

$V = \frac{2}{3} \pi \left(28\right)$

$V = \frac{56 \pi}{3}$