Question

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?

Answer 1

`color(blue)("Volume"=(56pi)/3)`

Explanation:

The volume of an ellipsoid is given by:

`V=4/3pi*a*b*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=4/3pir^3`

So volume of hemisphere is:

`V=2/3pir^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=4/3pi*a*b*c-2/3pir^3`

`V=2/3pi(2abc-r^3)`

Substituting values:

`V=2/3pi(2*3*3*2-2^3)`

`V=2/3pi(28)`

`V=(56pi)/3`