The problem is to identify the type of quadric surface represented by the equation $9x^2 + 25y^2 + 9z^2 = 225$.

1. Problem Description

The problem is to identify the type of quadric surface represented by the equation $9x^2 + 25y^2 + 9z^2 = 225$.

2. Solution Steps

First, we divide both sides of the equation by 225 to obtain a more standard form:

$\frac{9x^2}{225} + \frac{25y^2}{225} + \frac{9z^2}{225} = \frac{225}{225}$

$\frac{x^2}{25} + \frac{y^2}{9} + \frac{z^2}{25} = 1$

We can rewrite this as:

$\frac{x^2}{5^2} + \frac{y^2}{3^2} + \frac{z^2}{5^2} = 1$

This equation is of the form

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

where $a=5$, $b=3$, and $c=5$.

Since two of the denominators are equal ($a^2 = c^2 = 25$), this represents an ellipsoid of revolution. In particular, since $a=c > b$, it is an ellipsoid elongated along the y-axis. Also, it is an ellipsoid.

3. Final Answer

The quadric surface is an ellipsoid.