I need serious help..

Question

A satellite dish is shaped like a parabolic reflector. The satellite signals hit the surface of the dish and reflect to a single point, the focus, where the receiver is located. if the the dish is 4 feet across and 1 foot deep at its center, where should the receiver be located. What equation describes the cross section of this reflector?

Answer 1

Let's get an equation for the cross section first.  We'll place the origin at the lowest point in the reflector.  Another point on the reflector is (2,1), since from the center to the edge we move 2 units across, and the edge is 1 unit higher than the lowest point.

 

An equation for this parabola is y=ax², since it is vertically oriented and there is no translation.  To solve for a, we substitute (2,1) into the equation:  1 = a(2²), from which we find a = 1/4, and

 

y = x²/4.

 

An equation for a parabola which makes use of the distance from the focus to the vertex (lowest point on the reflector), where the parabola is vertically oriented and opens upwards (as it is in our setup) is

x² = 4cy.  Here, c is the distance from the vertex to the focus.  Rearranging the terms of our initial equation,

y = x²/4, we have

x² = 4y, from which we find that c = 1.  

 

The focus is 1 unit higher than the vertex.  The receiver should be placed at the focus, or at (0,1).