Chapter 8.4, Problem 1MRPS

Solution

(a)

To determine : The distance of the focus from the vertex.

The distance of the focus from the vertex is $8$ inches.

Given information :

Parabolic reflectors with a microphone at the focus allows to listen to sounds from far away. A parabolic microphone has a reflector that is $22.4$ inches in diameter and $8$ inches deep.

Explanation :

As the reflector is $8$ inches deep, it means that the focus lies $8$ inches above the vertex.

So, the focus of the parabolic microphone is $\left(8,0\right)$ .

Therefore, the distance of the focus from the vertex is $8$ inches.

(b)

To determine : An equation for the cross section of the reflector such that the vertex is at $\left(0,0\right)$ and the reflector opens to the right.

The required equation is $y^2=32x$ .

Given information :

Parabolic reflectors with a microphone at the focus allows to listen to sounds from far away. A parabolic microphone has a reflector that is $22.4$ inches in diameter and $8$ inches deep.

Formula used:

The standard equation of a parabola with vertex $\left(0,0\right)$ and focus at $\left(p,0\right)$ is given by,

  $y^2=4px$ .

Explanation :

The standard equation of a parabola with vertex $\left(0,0\right)$ and focus at $\left(p,0\right)$ is given by,

  $y^2=4px$ ....... (1)

The focus of the parabolic microphone is $\left(8,0\right)$ . So, substitute $p=8$ in equation (1).

  $\begin{array}{c}{y}^2=4\left(8\right)x\\ =32x\end{array}$

Therefore, the required equation is $y^2=32x$ .

(c)

To graph : The equation for the cross section of the reflector such that the vertex is at $\left(0,0\right)$ and the reflector opens to the right.

Given information :

Parabolic reflectors with a microphone at the focus allows to listen to sounds from far away. A parabolic microphone has a reflector that is $22.4$ inches in diameter and $8$ inches deep.

Graph :

The equation for the cross section of the reflector such that the vertex is at $\left(0,0\right)$ and the reflector opens to the right is $y^2=32x$ .

Graph the equation.

  

Interpretation :

The vertex is at $\left(0,0\right)$ and the focus is $\left(8,0\right)$ .