Question
problem 19
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 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.) x^2/100 + y^2/100 + z^2/601.92^2 = 1 In Part 1 of this module, you can move vertical and horizontal planes parallel to the coordinate planes through a quadric surface. The intersections of these planes with the surface form traces, which are shown both on the surface and on a 2D graph as you might draw on paper. You will be asked to investigate the families of traces in the exercises. Part 2 shows planes parallel to the coordinate planes and the traces of a surface they contain, but you will not see the surface itself. By moving the planes and watching the shapes of the traces, you can visualize the shape of the surface.
Explanation / Answer
The general equation is (x/a)^2 + (y/b)^2 - (z/c)^2 = 1
The radius at the center (z=0) is 100 m, so a=b=100
The radius at z=±500 is 130 m, so we have
(130/100)^2 - (500/c)^2 = 1
169/100 - 1 = (500/c)^2 ... add (500/c)^2 - 1
(69)/10 = 500/c ... square root
c = 500*10/69 = 5000/69
0
Substituting for c, your equation can be written as
(x^2 + y^2)/10000 - 69z^2/25000000 = 1
2500 x^2 + 2500 y^2 - 69 z^2 = 25000000
