A hyperbolic cooling tower is 100 meters wide at the base Th

A hyperbolic cooling tower is 100 meters wide at the base. The most narrow section of the tower is 100 meters above the ground and 40 meters wide. How wide is the cooling tower at its highest point, which is 160 meters above the ground?

Solution

Draw a sketch of a hyperbola with a base= 100 mt.Centre at ( 0,0)
Length of horizontal transverse axis=40

2a =40 ; a=20

right endpoint of base: ( 50 , - 100)

So, x^2/400 - y^2/b^2 = 1

2500/400 - 10000/b^2 =1 ;

25/4 -1 = 10000/b^2 ; 21/4 = 10000/b^2

b^2 = 40000/21

So,x^2/400 - y^2/(40000/21) = 1

Now find the (x , ( 160-40) ) = (x , 120) by plugging in   the equation :

x^2/400 - 120^2*21/40000 =1

x^2/400 - 14400*21/40000 =1

x^2 - 14400*21/100 = 400

x^2 - 144*21 = 400

x^2 = 400 +144*21

x =58.52 mt is the width of cooling tower at height 160mt above ground