Chapter 9.2, Problem 13CYU

(a)

To determine

To find: The focus is at the origin and write the equation for the parabola formed by the parabolic microphone.

Answer to Problem 13CYU

The

Explanation of Solution

Given:

The focus is 6 feet above the vertex and the latus rectum is 24 feet along.

Calculation:

Consider the general vertex form is,

  $y=a{\left(y-h\right)}^2+k$

The value of the latus rectum is 6 feet above the vertex and the latus rectum is 24 feet long. Then the value of a is,

  $\begin{array}{l}\frac{1}{a}=24\\ a=\frac{1}{24}\end{array}$

The focus of the parabola is at the origin, then then, the coordinates of the focus are,

  $\begin{array}{l}k+\frac{1}{4a}=0\\ k+\frac{1}{4\left(\frac{1}{24}\right)}=0\\ k=-6\end{array}$

The equation of the parabola is,

  $y=\frac{1}{24}{x}^2-6$

(b)

To determine

To find: The graph for the parabola.

Explanation of Solution

Given:

The focus is 6 feet above the vertex and the latus rectum is 24 feet along.

Calculation:

Consider the equation of the parabola is $y=\frac{1}{24}{x}^2-6$ .

Then, the graph of the parabola is shown in Figure 1

  

Figure 1