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Question

# Determine the value of $b$ so that the volume of the ellipsoid$x^2+\dfrac{y^2}{b^2}+\dfrac{z^2}{9}=1$is $16 \pi$.

Solution

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Answered 2 years ago
Answered 2 years ago
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One may be tempted to solve this via integration, but the task does not require us to do so. Instead, we can use the fact that the formula for the volume of the ellipsoid is:

$V=\dfrac{4\pi}{3}abc$

where the ellipsoid is given with the equation:

$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$

The equation we are given is:

$\dfrac{x^2}{1}+\dfrac{y^2}{b^2}+\dfrac{z^2}{9}=1$

From the equation we read $a=1$ and $c=3$. The volume has to be $16\pi$. Finally we have:

$\dfrac{4\pi}{3}\cdot 1\cdot b\cdot 3=16\pi$

Simplify the left side to get:

$4\pi b=16\pi\Rightarrow b=4$

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