Question

a parabolic microphone used on the sidelines of a professional football game uses a reflective dish 16 inch wide and 4 inch deep how far from the bottom of the disk should the microphone be placed?

Answer 1

The microphone should be placed 4 inches far from the bottom of the disk.

Explanation:

If we put the parabolic microphone into the coordinate system with the vertex at the origin and facing to the right, then the equation of the parabola would have the form:

`y^2=4ax` .... (1)

Where, a is the distance between focus and vertex.

The microphone to be placed at focus.

The parabolic microphone used on the sidelines of a professional football game uses a reflective dish 16 inch wide and 4 inch deep. It means the parabola passing through the points (4,8) and (4,-8).

Put x=4 and y=8 in equation (1).

`(8)^2=4a(4)`

`64=16a`

Divide both sides by 16.

`4=a`

Therefore, the microphone should be placed 4 inches far from the bottom of the disk.

Answer 2

If we put the parabola in a coordinate system with vertex at the origin and facing the positive x axis. We have the following points lying on the curve:
(4, 8), (4, -8)

For parabola facing right:
y2 = 4ax

Substituting either coordinates to solve for a:
a = 4

Therefore, the microphone should be placed 4 inches from the bottom of the disk.