An ellipsoid has radii with lengths of #6 #, #6 #, and #3 #. A portion the size of a hemisphere with a radius of #6 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Answer 1

Remaining volume is #0#

Volume of an ellipsoid with dimensions #a#, #b# and #c# is given by #4/3pixxaxxbxxc#, hence
volume of ellipsoid of radii with lengths of #6#, #6# and #3# is #4/3xxpixx6xx6xx3=144pi#.
Now as volume of sphere is given by #4/3pir^3#, volume of hemisphere of radius #6# cut from this is #1/2xx4/3pixx6^3=2/3xx216pi=144pi#
Hence, remaining volume of the ellipsoid is #144pi-144pi=0#
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Answer 2

To find the remaining volume of the ellipsoid after removing a hemisphere with a radius of 6, 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 given by:

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

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

Given that the ellipsoid has radii lengths of 6, 6, and 3, we have:

[ a = b = 6 ] [ c = 3 ]

The volume of the ellipsoid before removing the hemisphere is:

[ V_{\text{ellipsoid}} = \frac{4}{3} \pi (6)(6)(3) ]

Now, the volume of the removed hemisphere can be calculated using the formula:

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

Where ( r ) is the radius of the hemisphere, which is 6.

[ V_{\text{hemisphere}} = \frac{2}{3} \pi (6)^3 ]

The remaining volume of the ellipsoid after removing the hemisphere is:

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

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Answer from HIX Tutor

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.

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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