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 position of the hyperboloid is such that the center is at the origin with its axis along the z-axis, and the minimum diameter at the center.)

r² / 110² - 3z² / 1375² = 1

Explanation:

The equation of a hyperboloid (which is a hyperbola rotated about the z axis or conjugate axis) that is centered at the origin is:

x² / a² + y² / b² - z² / c² = 1

If the cross sections are circular rather than elliptical, then a = b.

(x² + y²) / a² - z² / c² = 1

Or, if you prefer cylindrical coordinates:

r² / a² - z² / c² = 1

We know that at z = 0, r = 110. And at z = - 500, r = 130.

110² / a² - 0 = 1

130² / a² - (-500) ² / c² = 1

Solving:

a² = 110²

c² = 1375² / 3

Plugging in:

r² / 110² - 3z² / 1375² = 1