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

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

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Answer 1 · 2018-06-06T15:42:56

$Volume = \frac{56 π}{3}$ $color(blue)("Volume"=(56pi)/3)$

Explanation:

The volume of an ellipsoid is given by:

$V = \frac{4}{3} π \cdot a \cdot b \cdot c$ $V=4/3pi*a*b*c$

Where #a, b, c are the radii of the ellipsoid.

The volume of a hemisphere is half the volume of a sphere:

Volume of a sphere is:

$V = \frac{4}{3} π r^{3}$ $V=4/3pir^3$

So volume of hemisphere is:

$V = \frac{2}{3} π r^{3}$ $V=2/3pir^3$

To find the volume of the ellipsoid when the hemisphere is removed, we just find the volume of the ellipsoid and subtract the volume of the hemisphere:

$V = \frac{4}{3} π \cdot a \cdot b \cdot c - \frac{2}{3} π r^{3}$ $V=4/3pi*a*b*c-2/3pir^3$

$V = \frac{2}{3} π (2 a b c - r^{3})$ $V=2/3pi(2abc-r^3)$

Substituting values:

$V = \frac{2}{3} π (2 \cdot 3 \cdot 3 \cdot 2 - 2^{3})$ $V=2/3pi(2*3*3*2-2^3)$

$V = \frac{2}{3} π (28)$ $V=2/3pi(28)$

$V = \frac{56 π}{3}$ $V=(56pi)/3$

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

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Explanation:

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An ellipsoid has radii with lengths of 33 , 33 , and 22 . A portion the size of a hemisphere with a radius of 22 is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

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