An ellipsoid has radii with lengths of $9$ , $11$ , and $4$ . A portion the size of a hemisphere with a radius of $8$ is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

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Answer 1 · 2016-03-12T19:42:19

$560/3 pi~~586.43$

The volume of an ellipsoid with radii $r_1$ , $r_2$ and $r_3$ is:

$4/3 pi r_1 r_2 r_3$

since it's basically the volume of a sphere stretched/squashed in a couple of dimensions.

The volume of a hemisphere of radius $r$ is:

$1/2*4/3 pi r^3 = 2/3 pi r^3$

In our example, $r_1 = 9$ , $r_2 = 11$ , $r_3 = 4$ and $r = 8$ so the volume remaining is:

$4/3 pi r_1 r_2 r_3 - 2/3 pi r^3$

$=2/3 pi (2*9*11*4 - 8^3)$

$=2/3 pi (792 - 512)$

$=2/3 pi * 280$

$=560/3 pi$

$~~586.43$

An ellipsoid has radii with lengths of 9 9 , 11 11 , and 4 4 . A portion the size of a hemisphere with a radius of 8 8 is removed form the ellipsoid. What is the remaining volume of the ellipsoid?