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?

In the example case:

This is the formula to find a hemisphere's volume:

In the example case:

When the hemisphere is removed from the ellipsoid, we must calculate its volume, which is;

Make the brackets simpler.

To find the remaining volume of the ellipsoid after removing a hemisphere-shaped portion, we can calculate the volume of the ellipsoid and subtract the volume of the removed portion.

1. Volume of the Ellipsoid: The formula for the volume of an ellipsoid is: [V_\text{ellipsoid} = \frac{4}{3} \pi abc] where (a), (b), and (c) are the semi-axes of the ellipsoid.

   Given the semi-axes lengths as 12, 11, and 8, we plug these values into the formula to find the initial volume of the ellipsoid.

2. Volume of the Removed Portion (Hemisphere): The volume of a hemisphere is given by the formula: [V_\text{hemisphere} = \frac{2}{3} \pi r^3] where (r) is the radius of the hemisphere.

   Given that the radius of the removed hemisphere is 9, we plug this value into the formula to find the volume of the removed portion.

3. Calculate Remaining Volume: Subtract the volume of the removed portion from the volume of the ellipsoid to find the remaining volume.

Now, perform these calculations to determine the remaining volume of the ellipsoid after removing the hemisphere-shaped portion.