An ellipsoid has radii with lengths of #6 #, #4 #, and #8 #. A portion the size of a hemisphere with a radius of #2 # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

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The volume that is still there is

The volume of the remaining ellipsoid after removing a hemisphere-shaped portion is calculated by subtracting the volume of the removed portion from the volume of the original ellipsoid.

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

Volume = (4/3) * π * a * b * c

Where 'a', 'b', and 'c' are the semi-axes of the ellipsoid.

Given that the semi-axes lengths of the ellipsoid are 6, 4, and 8, respectively, the volume of the original ellipsoid can be calculated as follows:

Volume_original = (4/3) * π * 6 * 4 * 8

Now, to find the volume of the hemisphere-shaped portion removed, we use the formula for the volume of a hemisphere:

Volume_hemisphere = (2/3) * π * r^3

Where 'r' is the radius of the hemisphere, given as 2.

Volume_hemisphere = (2/3) * π * 2^3

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

Volume_remaining = Volume_original - Volume_hemisphere

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

(V = \frac{4}{3}\pi a b c)

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

First, calculate the volume of the original ellipsoid:

(V_{\text{original}} = \frac{4}{3}\pi \times 6 \times 4 \times 8)

Next, calculate the volume of the removed hemisphere:

(V_{\text{hemisphere}} = \frac{2}{3}\pi r^3)

Where (r = 2).

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

(V_{\text{remaining}} = V_{\text{original}} - V_{\text{hemisphere}})