An ellipsoid has radii with lengths of $2$ $2$ , $5$ $5$ , and $3$ $3$ . A portion the size of a hemisphere with a radius of $1$ $1$ is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

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An ellipsoid has radii with lengths of $2$ $2$ , $5$ $5$ , and $3$ $3$ . A portion the size of a hemisphere with a radius of $1$ $1$ is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

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Answer 1 · 2016-10-29T14:58:27

Remaining volume of the ellipsoid is $\frac{118 π}{3}$ $(118pi)/3$

Explanation:

Volume of an ellipsoid with dimensions $a$ $a$ , $b$ $b$ and $c$ $c$ is given by $\frac{4}{3} π \times a \times b \times c$ $4/3pixxaxxbxxc$ , hence

volume of ellipsoid of radii with lengths of $2$ $2$ , $5$ $5$ and $3$ $3$ is $\frac{4}{3} π \times 2 \times 5 \times 3 = 40 π$ $4/3pixx2xx5xx3=40pi$ .

Now as volume of sphere is given by $\frac{4}{3} π r^{3}$ $4/3pir^3$ , volume of hemisphere of radius $1$ $1$ cut from this is $\frac{1}{2} \times \frac{4}{3} π \times 1^{3} = \frac{2}{3} \times π = \frac{2 π}{3}$ $1/2xx4/3pixx1^3=2/3xxpi=(2pi)/3$

Hence, remaining volume of the ellipsoid is $40 π - \frac{2 π}{3} = \frac{118 π}{3}$ $40pi-(2pi)/3=(118pi)/3$

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An ellipsoid has radii with lengths of $2$ $2$ , $5$ $5$ , and $3$ $3$ . A portion the size of a hemisphere with a radius of $1$ $1$ is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

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Explanation:

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An ellipsoid has radii with lengths of $2$ $2$ , $5$ $5$ , and $3$ $3$ . A portion the size of a hemisphere with a radius of $1$ $1$ is removed form the ellipsoid. What is the remaining volume of the ellipsoid?