Question

An ellipsoid has radii with lengths of `12`, `11`, and `8`. A portion the size of a hemisphere with a radius of `9` is removed from the ellipsoid. What is the remaining volume of the ellipsoid?

Answer 1

The remaining volume is `922pi` or `2897.71`.

Explanation:

The formula for the volume of an ellipsoid where the three radii are represented by `a`, `b`, and `c`, is:

`V_E=4/3piabc`

In the given case:

`V_E=4/3pixx12xx11xx8`

The formula for volume of a hemisphere is:

`V_H=2/3pir^3`

In the given case:

`V_H=2/3pixx9^3`

`V_H=2/3pixx729`

We need to determine the volume of the ellipsoid when the hemisphere is removed from it, which is;

`V_E-V_H=(4/3pixx12xx11xx8)-(2/3pixx729)`

Simplify the brackets.

`V_E-V_H=(4/cancel3pixx4cancel12xx11xx8)-(2/cancel3pixx243cancel729)`

`V_E-V_H=(4pixx4xx11xx8)-(2pixx243)`

`V_E-V_H=1408pi-486pi`

`V_E-V_H=(1408-486)pi`

`V_E-V_H=922pi`

Considering `pi` as `22/7`, we get:

`V_E-V_H=922xx22/7`

`V_E-V_H=2897.71`