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Find the volume of the ellipsoid x 2
+y 2
+7z 2
=25

User Saundra
by
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1 Answer

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The volume of the ellipsoid described by the equation
\( x^2 + y^2 + 7z^2 = 25 \) is approximately 197.90 cubic units.

To find the volume of an ellipsoid given by the equation
\(x^2 + y^2 + 7z^2 = 25\), we can recognize that this is the equation of an ellipsoid centered at the origin with semi-axes
\(a\),
\(b\), and
\(c\) in the
\(x\),
\(y\), and
\(z\) directions, respectively.

The general form of an ellipsoid's equation is:


\[(x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1\]

Comparing this with the given equation:


\[x^2 + y^2 + 7z^2 = 25\]

We can see that:


\[(x^2)/(5^2) + (y^2)/(5^2) + (z^2)/((√(25/7))^2) = 1\]

So, the semi-axes are:


\[a = b = 5, \quad \text{and} \quad c = \sqrt{(25)/(7)}\]

The volume \(V\) of an ellipsoid is given by the formula:


\[V = (4)/(3) \pi a b c\]

Let's substitute the values of \(a\), \(b\), and \(c\) into the formula to find the volume.

The volume of the ellipsoid described by the equation
\( x^2 + y^2 + 7z^2 = 25 \) is approximately 197.90 cubic units.

User ESDictor
by
8.7k points