A parabolic microphone used on the sidelines of a professional football game uses a reflective dish 24 inches wide and 5 inches deep. How far from the bottom of the disk should the microphone be placed?

Answer 1

Place the vertex at (0,0)
Recall that the parabola x^2 = 4py has its focus at (0,p)
Now, since y = ax^2, and y(12) = 5, a=5/144
so now we have y = 5/144 x^2
That is, x^2 = 144/5 y
That makes p = 144/20 = 36/5
The focus 36/5 inches from the vertex.