the optimal reception

Question

A Dish TV satellite dish is the shape of a paraboloid. The dish is 36 inches wide, and 8 inches deep. How many inches should the receiver be located from the vertex for optimal reception? (round to the nearest thousandth)

Answer 1

The standard form of the equation of  parabola with vertex at the origin is

 

  y = a x²    

 

   In this case y = 8  when x = 36/2 = 18 .  Plugging in and solving for a gives a = 0.02469

 

The P value for a parabola is  P =  1/( 4 a)  For this case P = 10.126

 

Since the distance from the vertex to the focal point is P,

 

The receiver should be located  10.126 inches from the vertex.