The crosssection of a nuclear power plants cooling tower is

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 252 meters and the diameter at its narrowest point 64 meters above the ground is 84 meters. If the diameter at the top of the tower is 168 meters, how tall is the tower? The tower is about meters tall. (Round to one decimal place as needed.)

Solution

Draw a sketch of a hyperbola with a base=252.
Length of horizontal transverse axis=84
Top=168
Set center at origin (0,0)
right endpoint of base: (126,-64)
Equation of hyperbola:x^2/a^2-y^2/b^2=1
given length of horizontal transverse axis=84=2a
a=42
a^2=1764
Using coordinates from endpoint of base, solve for b^2
Equation: 126^2/1764-64^2/b^2=1

15876/1764 - 4096/b^2 =1

9 - 4096/b^2 =1

9 -1 = 4096/b^2

b^2 = 4096/8

b^2 = 512

Equation: x^2/1764-y^2/512=1

plug x = 84 and solve for y

   84^2/1764 - y^2/512 =1

   4 - y^2/512 =1

   3 = y^2/512

   y^2 = 1536

   y = 39.2

the height of tower is 64+39.2 = 103.2 meters