Assume the parabola is placed on a graph where the x-axis is the top of the dish.
The vertex is then at (0,-30) The x-intercepts or zeros are at (-30,0) and (30,0)
The equation of such parabola would be:
[tex]y = a(x+30)(x-30)[/tex]
Plug in vertex to find value of 'a'
[tex]-30 = a(0+30)(0-30) \\ \\ a = \frac{-30}{(-30)(30)} = \frac{1}{30}[/tex]
Now find the focus given that [tex]p = \frac{1}{4a}[/tex]
[tex]p = \frac{1}{4(1/30)} = \frac{30}{4} = 7.5[/tex]
Answer: the microphone should be placed 7.5 inches from vertex.
