An ellipsoid has radii with lengths of $8$ $8$ , $9$ $9$ , and $2$ $2$ . A portion the size of a hemisphere with a radius of $3$ $3$ is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

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An ellipsoid has radii with lengths of $8$ $8$ , $9$ $9$ , and $2$ $2$ . A portion the size of a hemisphere with a radius of $3$ $3$ is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

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Answer 1 · 2017-04-07T18:14:29

$174 π \approx 546.637 {units}^{3}$ $174 pi ~~ 546.637 " units"^3$

Explanation:

Volume of an ellipsoid $V_{e l l i p s o i d} = \frac{4}{3} π a b c$ $V_(ellipsoid) = 4/3 pi abc$ where $a, b, c$ $a, b, c$ are the radii.

Volume of a hemisphere is half the volume of a sphere $V_{h e m i s p h e r e} = \frac{1}{2} \cdot \frac{4}{3} π r^{3} = \frac{2}{3} π r^{3}$ $V_(hemisphere) = 1/2*4/3 pi r^3 = 2/3 pi r^3$

$V_{e l l i p s o i d} - V_{h e m i s p h e r e} = \frac{4}{3} π \cdot 8 \cdot 9 \cdot 2 - \frac{2}{3} π 3^{3}$ $V_(ellipsoid) - V_(hemisphere) = 4/3 pi *8*9*2 - 2/3 pi 3^3$

$= 192 π - 18 π = 174 π$ $= 192 pi - 18 pi = 174 pi$

$V_{e l l i p s o i d} - V_{h e m i s p h e r e} = 174 π \approx 546.637 {units}^{3}$ $V_(ellipsoid) - V_(hemisphere) = 174 pi ~~ 546.637 " units"^3$

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An ellipsoid has radii with lengths of $8$ $8$ , $9$ $9$ , and $2$ $2$ . A portion the size of a hemisphere with a radius of $3$ $3$ is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

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An ellipsoid has radii with lengths of $8$ $8$ , $9$ $9$ , and $2$ $2$ . A portion the size of a hemisphere with a radius of $3$ $3$ is removed form the ellipsoid. What is the volume of the remaining ellipsoid?