Question

College Calculus III:

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 300 m and the minimum diameter, 500 m above the base, is 200 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.)

Answer 1

Here's the example I have; it doesn't really explain how to do it well.

Answer 2

Heres a graph in 2-D


Each cross-section is a circle, which means if you plug in a z-value then on the xy plane you have a circle.
\[x^2 + y^2 = r^2\]
We are given the radius of 2 circles at the points z=0 (radius is 100)
and z=500 (radius is 150)
This will allow you to solve for the unknown coefficients a,c in the hyperboloid equation:
\[\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1\]
Put it in circle equation form (isolate x^2 + y^2)
\[x^2 + y^2 = a^2(1+ \frac{z^2}{c^2}) = r^2\]
First plug in z=0 to solve for "a".
Then plug in z= 500 to solve for c