Chapter 10, Problem 13CT

To determine

Where should the microphone be placed, if a parabolic reflector (paraboloid of revolution) is used by the TV crews at the football game to pick up the referee’s announcements, quarterback signals, and so on. A microphone is placed at the focus of the parabola, if a certain reflector is $4$ feet wide and $1.5$ feet deep.

Answer to Problem 13CT

Solution:

The microphone should be placed $\frac{2}{3}$feet from the base of the reflector, along its axis of symmetry.

Explanation of Solution

Given information:

A parabolic reflector (paraboloid of revolution) is used by the TV crews at the football game to pick up the referee’s announcements, quarterback signals, and so on. A microphone is placed at the focus of the parabola, if a certain reflector is $4$ feet wide and $1.5$ feet deep.

Draw the parabola used to form the reflector which is $4$ feet wide and $1.5$ feet deep on a rectangular coordinate system, so that the vertex of the parabola is at the origin and its focus is on the positive $y-$ axis.

  

Thus, the general form of the equation of the parabola is $x^2=4ay$ and its focus is at $(0,a)$ .

Since the point $(2,1.5)$ is on the graph,

To find the focus, substitute $x=2$ and $y=1.5$ in $x^2=4ay$

  $\Rightarrow 2^2=4a(1.5)$

  $\Rightarrow 4=6a$

  $\Rightarrow a=\frac{2}{3}$

The microphone should be placed $\frac{2}{3}$ feet from the base of the reflector, along its axis of symmetry.

Following graph shows the location of the microphone $F=\left(0,\frac{2}{3}\right)$,