An ellipsoid has radii with lengths of #9 #, #12 #, and #10 #. 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?

The ellipsoid's volume is

The hemisphere's volume is

The volume that is still there is

To find the remaining volume of the ellipsoid after removing a portion the size of a hemisphere with a radius of 5, we can first calculate the volume of the original ellipsoid and then subtract the volume of the removed portion.

The formula for the volume of an ellipsoid is given by:

[ V = \frac{4}{3} \pi a b c ]

where (a), (b), and (c) are the semi-axes of the ellipsoid.

For the original ellipsoid with semi-axes lengths of 9, 12, and 10, we have:

[ a = 9, , b = 12, , c = 10 ]

Now, let's calculate the volume of the original ellipsoid:

[ V_{\text{original}} = \frac{4}{3} \pi (9)(12)(10) ]

Next, we need to find the volume of the portion that is removed, which is a hemisphere with a radius of 5. The formula for the volume of a hemisphere is:

[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 ]

where (r) is the radius of the hemisphere.

For the removed hemisphere, (r = 5), so:

[ V_{\text{removed}} = \frac{2}{3} \pi (5^3) ]

Now, to find the remaining volume of the ellipsoid, we subtract the volume of the removed portion from the volume of the original ellipsoid:

[ V_{\text{remaining}} = V_{\text{original}} - V_{\text{removed}} ]

Substituting the values:

[ V_{\text{remaining}} = \frac{4}{3} \pi (9)(12)(10) - \frac{2}{3} \pi (5^3) ]

[ V_{\text{remaining}} = \frac{4}{3} \pi (1080) - \frac{2}{3} \pi (125) ]

[ V_{\text{remaining}} = \frac{4320}{3} \pi - \frac{250}{3} \pi ]

[ V_{\text{remaining}} = \left( \frac{4320 - 250}{3} \right) \pi ]

[ V_{\text{remaining}} = \left( \frac{4070}{3} \right) \pi ]

So, the remaining volume of the ellipsoid after removing the hemisphere portion is ( \frac{4070}{3} \pi ).