An ellipsoid has radii with lengths of #4 #, #5 #, and #7 #. 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 volume of the ellipsoid is

The hemisphere's volume is

The amount of volume left is

The volume of an ellipsoid with radii (a), (b), and (c) is given by the formula:

[ V_{\text{ellipsoid}} = \frac{4}{3}\pi abc ]

For the given ellipsoid with radii (a = 4), (b = 5), and (c = 7), the volume is:

[ V_{\text{ellipsoid}} = \frac{4}{3}\pi (4)(5)(7) = \frac{4}{3}\pi (140) = \frac{560}{3}\pi ]

The volume of a sphere is given by:

[ V_{\text{sphere}} = \frac{4}{3}\pi r^3 ]

The volume of a hemisphere (half of a sphere) with radius (r = 5) is half of the sphere's volume, so:

[ V_{\text{hemisphere}} = \frac{1}{2} \left( \frac{4}{3}\pi (5)^3 \right) = \frac{1}{2} \left( \frac{4}{3}\pi (125) \right) = \frac{250}{3}\pi ]

To find the remaining volume of the ellipsoid after the hemisphere has been removed, subtract the volume of the hemisphere from the volume of the ellipsoid:

[ V_{\text{remaining}} = V_{\text{ellipsoid}} - V_{\text{hemisphere}} = \frac{560}{3}\pi - \frac{250}{3}\pi = \frac{310}{3}\pi ]

Therefore, the remaining volume of the ellipsoid, after removing a portion the size of a hemisphere with radius 5, is ( \frac{310}{3}\pi ) cubic units.