Chapter 8.2, Problem 57PS

Solution

(a)

To graph : The antenna twice: once opening upward and one opening left.

Given information :

The parabolic antenna used by a television station to transmit is $146$ inches in diameter. Its focus is $48$ inches from the vertex.

Graph :

The focus of the antenna opening upward lies $48$ units above the vertex. So, the graph will be,

  

Here, the focus is $\left(0,48\right)$ and the directrix is $y=-48$ .

The focus of the antenna opening left lies $48$ units left from the vertex. So, the graph will be,

  

Here, the focus is $\left(-48,0\right)$ and the directrix is $x=48$ .

Interpretation :

The focus of the graph opening upward is $\left(0,48\right)$ and the directrix is $y=-48$ and the focus of the graph opening left is $\left(-48,0\right)$ and the directrix is $x=48$ .

(b)

To determine : Two equations for the antenna's cross section: one of the form $x^2=4py$ and one of the form $y^2=4xy$ on the basis of the graphs in part (a).

The two equations for the antenna's cross section are $x^2=192y$ and $y^2=-192x$ .

Given information :

The parabolic antenna used by a television station to transmit is $146$ inches in diameter. Its focus is $48$ inches from the vertex.

Formula used:

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

  $x^2=4py$

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

  $y^2=4px$

Explanation :

Consider the standard equation $x^2=4py$ .

Substitute $p=48$ , so

  $\begin{array}{c}{x}^2=4\left(48\right)y\\ =192y\end{array}$

Consider the standard equation $y^2=4px$ .

Substitute $p=-48$ , so

  $\begin{array}{c}{y}^2=4\left(-48\right)x\\ =-192x\end{array}$

Therefore, the two equations for the antenna's cross section are $x^2=192y$ and $y^2=-192x$ .

(c)

To calculate : The depth of the antenna's dish.

The depth of the antenna is $27.75\text{ inches}$ .

Given information :

The parabolic antenna used by a television station to transmit is $146$ inches in diameter. Its focus is $48$ inches from the vertex.

Calculation :

The diameter is $146$ inches, this means the antenna extends $\frac{146}{2}$ inches to both sides of the vertex $\left(0,0\right)$ .

Substitute $x=\frac{146}{2}\text{ or }73$ in $x^2=192y$ .

  $\begin{array}{c}{\left(73\right)}^2=192y\\ y=\frac{5329}{192}\\ \approx 27.75\text{ inches}\end{array}$

Now, substitute $y=73$ in $y^2=-192x$ .

  $\begin{array}{c}{\left(73\right)}^2=-192x\\ x=-\frac{5329}{192}\\ \approx -27.75\text{ inches}\end{array}$

The depth is same by using both the equations, the difference is of the positive and negative sign.

Therefore, the depth of the antenna is $27.75\text{ inches}$ .