Chapter 12.4, Problem 44STP

To determine

Write the equation for the cross-section.

Answer to Problem 44STP

  $y=\frac{1}{24}{\left(x+6\right)}^2+0$ .

Explanation of Solution

Given information:

A microphone is placed at the focus of parabolic reflector to collect sound for the television broadcast of a football game. Focus is at origin, the focus is $6inches$ from the vertex and the parabola opens to the right.

Calculation:

The vertex equation for the parabola is,

  $\begin{array}{l}y=a{\left(x-h\right)}^2+k\\ where\left(h,k\right)isthevertex\\ focusisatorigin\left(0,0\right)\end{array}$

The parabola is opens to the right, we know that the vertex is $6inches$ to the left of the focus.

Therefore the vertex is at $\left(-6,0\right)$

In order to find the value the length of the focus is equal to $1/a$ .

  $\begin{array}{l}1/a=24,a=1/24\\ sotheequationforparabolais,\\ y=\frac{1}{24}{\left(x-\left(-6\right)\right)}^2+0\\ y=\frac{1}{24}{\left(x+6\right)}^2+0\end{array}$

Hence, the equation of parabola is $y=\frac{1}{24}{\left(x+6\right)}^2+0$ .