The volume of the ellipsoid described by the equation
is approximately 197.90 cubic units.
To find the volume of an ellipsoid given by the equation
, we can recognize that this is the equation of an ellipsoid centered at the origin with semi-axes
,
, and
in the
,
, and
directions, respectively.
The general form of an ellipsoid's equation is:
![\[(x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/84dexxb2o1qve4wmgjtww3afiipm9xsbwm.png)
Comparing this with the given equation:
![\[x^2 + y^2 + 7z^2 = 25\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/6vcg53vxsse7s4c0x1g5ptoa8uk31t567p.png)
We can see that:
![\[(x^2)/(5^2) + (y^2)/(5^2) + (z^2)/((√(25/7))^2) = 1\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ax62iwjimn46fm7xpk9hj7z1bu99x84wt9.png)
So, the semi-axes are:
![\[a = b = 5, \quad \text{and} \quad c = \sqrt{(25)/(7)}\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/bci3gnedx32uzjt4c1jstuqfmhm4olji93.png)
The volume \(V\) of an ellipsoid is given by the formula:
![\[V = (4)/(3) \pi a b c\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/tp18u84l5wqldqfhgrclqpt2b9wu9k9rec.png)
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
is approximately 197.90 cubic units.