An ellipsoid has radii with lengths of #14 #, #7 #, and #6 #. A portion the size of a hemisphere with a radius of #9 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

To find the remaining volume of the ellipsoid after removing a portion the size of a hemisphere, we first calculate the volume of the ellipsoid using the formula:

[ V_{\text{ellipsoid}} = \frac{4}{3} \pi a b c ]

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

Given that the semi-axes of the ellipsoid are ( 14 ), ( 7 ), and ( 6 ), respectively, we have:

[ a = 14, \quad b = 7, \quad c = 6 ]

Substituting these values into the formula:

[ V_{\text{ellipsoid}} = \frac{4}{3} \pi (14)(7)(6) ]

[ V_{\text{ellipsoid}} = \frac{4}{3} \pi (588) ]

[ V_{\text{ellipsoid}} = 784 \pi ]

The volume of the removed portion, which is a hemisphere, can be calculated using the formula for the volume of a hemisphere:

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

Given that the radius of the hemisphere is ( 9 ), we have:

[ r = 9 ]

Substituting this value into the formula:

[ V_{\text{hemisphere}} = \frac{2}{3} \pi (9)^3 ]

[ V_{\text{hemisphere}} = \frac{2}{3} \pi (729) ]

[ V_{\text{hemisphere}} = 486 \pi ]

To find the remaining volume of the ellipsoid, we subtract the volume of the hemisphere from the volume of the ellipsoid:

[ V_{\text{remaining}} = V_{\text{ellipsoid}} - V_{\text{hemisphere}} ]

[ V_{\text{remaining}} = 784 \pi - 486 \pi ]

[ V_{\text{remaining}} = 298 \pi ]

Therefore, the remaining volume of the ellipsoid after removing a portion the size of a hemisphere with a radius of ( 9 ) is ( 298 \pi ).