Parabolic Arcs

Question

A parabolic reflector is in the shape made by revolving an arc of a parabola, starting at the vertex, about the axis of the parabola. If the focus is 9 inches from the vertex, and the parabolic arc is 16 inches deep, how wide is teh opening of the reflector?

Answer 1

The general formula for a parabola with a vertical axis is y-k = 4p(x-h)².

 

p is the directed distance from the vertex to the focus, which in your case would be 9. Let's assume the origin is he vertex and the focus would then have the coordinates (0,9).

 

So your equation would be y-0 = 36(x-0)² or y=36x².

 

So when you pick an x value, you get a corresponding which corresponds to the height above the x-axis. So if the parabolic arc is 16 inches deep, we need to find what x leads to 16 or sove

 

16=36x²

16/36 = x²

x= + or - 4/6

 

So the opening of the reflector goes from -2/3 to 2/3 so the dish is 4/3 inches wide.

 

Hope this helped.