Hyperbola Application Problem

Question

The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 270 feet. The smallest diameter of the tower is 148 which is 416 feet above the ground. The tower is 538 feet tall. Find the width of the tower at a height of 120 feet, to 1 decimal place.

Answer 1

The general form of a hyperbola with transverse (intersects the hyperbola) axis parallel to the x-axis is ((x-h)²⁄a²) - ((y-k)²⁄b²) = 1, where (h,k) are the coordinates of the center of the hyperbola. The narrowest part of the hyperbola is the origin.

In this case, the narrowest part of the tower occurs at a height of 416ft. Therefor k = 416ft For convenience, the center of the tower / conjugate axis will be defined as the y-axis. This makes h = 0.

The distance between the vertices of a hyperbola - which are the intersection of the transverse axis and the hyperbola - is 2a = 148ft ==> a = 148/2 = 74ft

To find the value of b, you need the x and y values of some other part of the tower. The only other point where all values are known is at the base of the tower. The diameter of the base = 2x_base = 270ft, so x at the base of the tower is 270/2 = 135ft and the base of the tower has a height y = 0

THis gives the equation: ((x-h)²⁄a²) - ((y-k)²⁄b²) = 1

((135)²⁄ 74²) - ((0-416)²⁄b²) = 1, solve for b. b = √(416²) ⁄ (((135)²⁄ 74²)-1) ≅ 272.6ft

The final equation is: (x²⁄ 74²) - ((y-416)²⁄ 272.6²) = 1

Given any other value of x or y on the hyperbola, you can solve for the other value by rearranging the equation. In the case of this problem, you are given a height, or y value, so you will be solving for x, which is half the width of the tower at that height. Recommend you try to solve the problem from here and check with the answer below.

x = 74√(1 + ((120-416)²⁄ 272.6²)) = 109.2ft, width of tower is 2x = 218,5ft

LOGIC CHECK: The narrowest part of the tower is at 416ft and the base is the widest part since it is farthest from the center of the hyperbola. Since the point in question is between these two, the answer should be between the largest and smallest values: 148ft < 218.4ft < 270ft. NOTE: a logic check is always a good idea. If the result doesn't make logical sense, then it is probably wrong. That doesn't mean that any answer that makes logical sense is necessarily correct, but the odds are higher if it is logically consistent with what is expected.