Question

The​ cross-section of a nuclear power​ plant's cooling tower is in the shape of a hyperbola. Suppose the tower has a base diameter of

156 meters and the diameter at its narrowest​ point, 48 meters above the​ ground, is 52 meters. If the diameter at the top of the tower is 104

​meters, how tall is the​ tower?

The tower is about ___ meters tall.

Answer 1

The general equation of the Hyperbola

where (h,k) is the coordinate of the center of hyperbola

assuming that the y axis of our coordinate system runs vertically through the center of the tower, The coordinates of the center of hyperbola are h=0,k=48

The equation of hyperbola is then,

from the question, we know two points on the curve of hyperbola

since the diameter is 156 m at ground level and y axis as the middle of the tower, The point x is 156/2=78 .

also given diameter at 48m above the ground is 52m

so the (52/2,48) is also a point on hyperbola. (52/2 is taken because y axis as middle)

point is (26,48)

substitute x=26,y=48 in equation

so

updating the equation with then

substituting the other known point (78,0) in the equation

  

again update the equation with

given the diameter at the top of the tower is 104

so the x coordinate =104/2=52.

to get the height of the tower put x=52 and solve for y

taking square root on both sides

so height of the tower is 77.3939 m