# An ellipsoid has radii with lengths of #5 #, #12 #, and #8 #. A portion the size of a hemisphere with a radius of #7 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Thus, the Ellipsoid's residual volume

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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 removed hemisphere.

The formula for the volume of an ellipsoid is:

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

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

Given that the radii lengths of the ellipsoid are (a = 5), (b = 12), and (c = 8), the volume of the original ellipsoid is:

[ V_{\text{original}} = \frac{4}{3} \pi (5)(12)(8) ]

To find the volume of the hemisphere that's removed, we use the formula for the volume of a hemisphere:

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

Where (r) is the radius of the hemisphere, given as (r = 7).

Substituting the values, we get:

[ V_{\text{hemisphere}} = \frac{2}{3} \pi (7)^3 ]

Now, to find the remaining volume, subtract the volume of the hemisphere from the original volume of the ellipsoid:

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

[ V_{\text{remaining}} = \frac{4}{3} \pi (5)(12)(8) - \frac{2}{3} \pi (7)^3 ]

Calculate the values and you'll get the remaining volume of the ellipsoid.

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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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