# An ellipsoid has radii with lengths of 12 , 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?

Apr 25, 2017

The remaining volume is $700 \frac{2}{3} \pi$ or $2200.09$.

#### Explanation:

The method to determine the remaining volume of the ellipsoid is to subtract the volume of the hemisphere from the original 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, and $a$, $b$, and $c$ are the radii of the ellipsoid.

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

Hence, the remaining volume can be calculated as ${V}_{E} - {V}_{H}$:

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

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

${V}_{E} - {V}_{H} = \frac{2}{3} \pi \left(\left[2 \times 12 \times 7 \times 7\right] - \left[{5}^{3}\right]\right)$

${V}_{E} - {V}_{H} = \frac{2}{3} \pi \left(1176 - 125\right)$

${V}_{E} - {V}_{H} = \frac{2}{3} \pi \times 1051$

${V}_{E} - {V}_{H} = \frac{2102}{3} \pi = 700 \frac{2}{3} \pi$

Considering $\pi$ as $3.14$:

${V}_{E} - {V}_{H} = \frac{2102}{3} \times 3.14$

${V}_{E} - {V}_{H} = 2200.09$