Cooling towers for nuclear reactors are often constructed as hyperboloids of one sheet because of the structural stability of that surface. Suppose all horizontal cross sections are circular, with a minimum radius of 200 feet occurring at a height of 600 feet. The tower is to be 800 feet tall with a maximum cross-sectional radius of 300 feet. Find the equation of the surface.

Make the axis of symmetry the y axis and the minimum cross section be the y = 0 plane. The general equation for the hyperbola that generates the surface is

x^2/a^2 - y^2/b^2 = 1

where a is the minimum radial distance from the y axis, which in this case is 200 feet. To get b, require that x = 300 when y = 200 feet. (That is, y = 200 feet above the "throat" where x = 200.
(300/200)^2 - (200/b)^2 = 1
(200/b)^2 = 1.25
b/200 = sqrt(4/5)
b = 178.9 ft

(x/200)^2 - (y/178.9)^2 = 1
The domain of y is -600 to +200

However, the equation is for a hyperboloid of one sheet, not a hyperbola. I need to find a equation that follows the general equation of a hyperboloid of one sheet: ax^2 + by^2 - cz^2 where a,b,c are > 0.

The hyperbola that I described, when rotated about the y axis, in an x,y,z coordinate system. becomes the hyperboloid of one sheet. I leave the rest up to you

To find the equation of the surface for the cooling tower, we can use the equation for a hyperboloid of one sheet given that all horizontal cross-sections are circular.

The equation for a hyperboloid of one sheet in standard form is:

(x^2 / a^2) - (y^2 / b^2) - (z^2 / c^2) = 1

Where a, b, and c are the constants determining the shape and size of the hyperboloid.

In this case, since all horizontal cross-sections are circular, we need to find the values of a, b, and c to match the given conditions.

Let's analyze the given information step by step:

1. The minimum radius of the circular cross-section occurs at a height of 600 feet and has a radius of 200 feet. This corresponds to a point on the surface of the hyperboloid, which we can denote as (x1, y1, z1) = (0, 200, 600).

2. The maximum radius of the circular cross-section occurs at the top of the tower, which has a radius of 300 feet. This corresponds to another point on the surface of the hyperboloid, which we can denote as (x2, y2, z2) = (0, 300, 800).

Now, we can find the values of a, b, and c using the given points:

First, let's find c:
The value of c is simply the difference in the z-coordinates of the two points on the hyperboloid.
c = z2 - z1 = 800 - 600 = 200

Next, let's find a and b:
Since all horizontal cross-sections are circular, the variation in the radii is along the y-axis. Thus, we can consider the points (0, 200) and (0, 300) as points on an ellipse in the yz-plane.

To find a and b, we need to determine the equation of this ellipse. Since it is an ellipse centered at the origin in the yz-plane, the equation for the ellipse becomes:

(y^2 / b^2) + (z^2 / c^2) = 1

Using the point (0, 200, 600), we can substitute y = 200 and z = 600 into the equation:
(200^2 / b^2) + (600^2 / c^2) = 1

Simplifying:
40000 / b^2 + 360000 / c^2 = 1

Using the point (0, 300, 800), we can substitute y = 300 and z = 800 into the equation:
(300^2 / b^2) + (800^2 / c^2) = 1

Simplifying:
90000 / b^2 + 640000 / c^2 = 1

Now, we have a system of two equations:
40000 / b^2 + 360000 / c^2 = 1
90000 / b^2 + 640000 / c^2 = 1

We solve this system of equations to find the values of b and c.

By rearranging the first equation, we get:
40000 / b^2 = 1 - 360000 / c^2

Substituting this equation into the second equation:
90000 / b^2 + 640000 / c^2 = 1
90000 /(1 - 360000 / c^2) + 640000 / c^2 = 1

Simplifying the equation above, we can solve for c^2:
90000 * c^2 + 640000 * (1 - 360000 / c^2) = c^2

90000 * c^2 + 640000 - 230400000000 / c^2 = c^2

Rearranging the equation:
90000 * c^4 + 640000 * c^2 - 230400000000 = 0

Now, we have a quadratic equation in terms of c^2, which we can solve to find c.

Once we have the values of a, b, and c, we can substitute them into the equation for the hyperboloid:

(x^2 / a^2) - (y^2 / b^2) - (z^2 / c^2) = 1

to obtain the equation of the surface for the cooling tower.