# An ellipsoid has radii with lengths of 8 , 6 , and 4 . A portion the size of a hemisphere with a radius of 2  is removed from the ellipsoid. What is the volume of the remaining ellipsoid?

Jan 28, 2018

$250 \frac{2}{3} \pi = V$

#### Explanation:

Remember that the formula for the volume of an ellipsoid is:
$V = \frac{4}{3} \pi \left({r}_{1} \cdot {r}_{2} \cdot {r}_{3}\right)$

We plug the given radii lengths to find the volume.

=>$V = \frac{4}{3} \pi \left(8 \cdot 6 \cdot 4\right)$
=>$V = \frac{4}{3} \pi 192$
=>$V = 256 \pi$

Now, a hemisphere with a radius of 2 is removed from the ellipsoid.
We have to subtract the volume of the hemisphere from the volume of the ellipsoid.

A volume of a hemisphere is: $\frac{1}{2} \cdot \frac{4}{3} \cdot \pi {r}^{3}$ We now solve for the volume.
=>$V = \frac{1}{2} \cdot \frac{4}{3} \cdot \pi {r}^{3}$
=>$V = \frac{4}{6} \cdot \pi {\left(2\right)}^{3}$
=>$V = \frac{4}{6} \cdot \pi 8$
=>$V = \frac{16}{3} \cdot \pi$ We now subtract.

=>$256 \pi - \frac{16}{3} \pi = V$

=>$\pi \left(256 - \frac{16}{3}\right) = V$
=>$\pi \left(\frac{752}{3}\right) = V$
=>$250 \frac{2}{3} \pi = V$