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

Apr 24, 2017

The remaining volume is $\frac{730}{3} \pi$ or $764.06$.

#### Explanation:

To determine the remaining volume of the ellipsoid, we need to subtract the volume of the hemisphere from the volume of the ellipsoid.

The formula for volume of an ellipsoid is:
${V}_{E} = \frac{4}{3} \pi a b c$, where ${V}_{E} =$Volume of ellipsoid, $\pi = 3.14$, and $a$, $b$, and $c$ are the radii.

The formula for volume of a hemisphere is:
${V}_{H} = \frac{2}{3} \pi {r}^{3}$, where ${V}_{H} =$Volume of hemisphere, $\pi = 3.14$, and $r =$radius.

Hence the remaining volume of the ellipsoid will be:

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

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

${V}_{E} - {V}_{H} = \frac{2}{3} \pi \left(\left[2 \times 5 \times 7 \times 7\right] - \left[{5}^{3}\right]\right)$

${V}_{E} - {V}_{H} = \frac{2}{3} \pi \left(490 - 125\right)$

${V}_{E} - {V}_{H} = \frac{2}{3} \pi \times 365$

${V}_{E} - {V}_{H} = \frac{730}{3} \pi$

${V}_{E} - {V}_{H} = \frac{730}{3} \times 3.14$

${V}_{E} - {V}_{H} = 764.06$