# An ellipsoid has radii with lengths of #3 #, #4 #, and #2 #. A portion the size of a hemisphere with a radius of #2 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Remaining volume of the ellipsoid is

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To find the remaining volume of the ellipsoid after removing a hemisphere with a radius of 2, we first calculate the volume of the ellipsoid and then subtract the volume of the hemisphere.

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

[ 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 lengths of the radii of the ellipsoid are 3, 4, and 2, we have:

( a = 3 ), ( b = 4 ), and ( c = 2 ).

Substitute these values into the formula to find the volume of the ellipsoid:

[ V_{\text{ellipsoid}} = \frac{4}{3} \pi (3)(4)(2) = \frac{32}{3} \pi ]

Next, we calculate the volume of the hemisphere using the formula:

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

where ( r = 2 ) (radius of the hemisphere).

Substitute ( r = 2 ) into the formula to find the volume of the hemisphere:

[ V_{\text{hemisphere}} = \frac{2}{3} \pi (2)^3 = \frac{16}{3} \pi ]

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

[ \text{Remaining volume} = V_{\text{ellipsoid}} - V_{\text{hemisphere}} ] [ = \frac{32}{3} \pi - \frac{16}{3} \pi ] [ = \frac{16}{3} \pi ]

So, the remaining volume of the ellipsoid after removing a hemisphere with a radius of 2 is ( \frac{16}{3} \pi ) 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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