I have one question left in an assignment and I havn't been able to solve it. I know the equaton for a hyperboloid and I know that $a$ and $b$ will be equal to each other. I don't know how to solve for $c$ though. $a = b = 110$ in this problem.
A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is $260$ m and the minimum diameter, $500$ m above the base, is $220$ m. Find an equation for the tower. (Assume the center is at the origin with axis the z-axis and the minimum diameter is at the center.)
It's convenient to put the origin of coordinate system in the center of the narrowest part. Then the equation takes the form $$ \frac{x^2}{a^2}+\frac{y^2}{a^2} = 1 + \frac{z^2}{c^2} $$ where $a=110$ as you noted. At vertical distance $z$ from the center, the radius of cross-section is $$ R = a\sqrt{1+z^2/c^2} $$ It remains to plug $z=500$, equate the radius to $260/2=130$, and solve for $c$.