# An ellipsoid has radii with lengths of #8 #, #7 #, 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 remaining volume is

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To find the remaining volume of the ellipsoid after removing a hemisphere, you can use the formula for the volume of an ellipsoid and subtract the volume of the hemisphere.

The formula for the volume ( V ) of an ellipsoid with semi-axes ( a ), ( b ), and ( c ) is given by:

[ V = \frac{4}{3} \pi abc ]

Given the semi-axes lengths of the ellipsoid as ( a = 8 ), ( b = 7 ), and ( c = 7 ), we have:

[ V_{\text{ellipsoid}} = \frac{4}{3} \pi (8)(7)(7) ]

Now, let's calculate the volume of the removed hemisphere. The formula for the volume ( V_h ) of a hemisphere with radius ( r ) is:

[ V_h = \frac{2}{3} \pi r^3 ]

Given ( r = 5 ), we have:

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

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

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

[ V_{\text{remaining}} = \left(\frac{4}{3} \pi (8)(7)(7)\right) - \left(\frac{2}{3} \pi (5)^3\right) ]

[ V_{\text{remaining}} = \frac{4}{3} \pi (8)(7)(7) - \frac{2}{3} \pi (5)^3 ]

[ V_{\text{remaining}} \approx 1054.73 ]

So, the remaining volume of the ellipsoid after removing the hemisphere is approximately ( 1054.73 ) cubic units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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