Question

A parabolic microphone used on the sidelines of a professional football game uses a reflective dish 24 in. wide and 5 in. deep. How far from the bottom of the dish should the microphone b placed?

Answer 1

check the picture below.

so the microphone looks more or less like that.

notice, if we put the center of it at the origin, the vertex (h,k) = (0,0), and the parabola passes through (12, 5). How far from the bottom? well, that'd be the focus point of the parabola, and that'd be "p" distance from the vertex.

`\bf \begin{array}ll \cline{1-1} ~\hfill \textit{parabola vertex form}~\hfill \\\\ \begin{array}{llll} y=a(x- h)^2+ k\\\\ x=a(y- k)^2+ h \end{array} \qquad\qquad vertex~~(\stackrel{0}{ h},\stackrel{0}{ k}) \\\\ \cline{1-1} \end{array}\qquad \qquad y=a(x-0)^2+0 \\\\\\ \textit{we also know that } \begin{cases} x=12\\ y=5 \end{cases}\implies 5=a(12-0)^2+0\implies 5=144a`

`\bf \cfrac{5}{144}=a\qquad therefore\qquad y=\cfrac{5}{144}x^2\impliedby \textit{let's put that in vertex/focus form} \\\\\\ 144y=5x^2\implies \cfrac{144}{5}y=x^2\implies \stackrel{\stackrel{4p}{\downarrow }}{\cfrac{144}{5}}(y-0)=(x-0)^2 \\\\[-0.35em] ~\dotfill\\\\ 4p=\cfrac{144}{5}\implies p=\cfrac{144}{20}\implies \blacktriangleright p=7.2 \blacktriangleleft`

Answer 2

answer : C 7.2 in

A parabolic microphone used on the sidelines of a professional football game uses a reflective dish 24 in. wide and 5 in. deep.

The parabolic microphone is placed horizontally

so we use equation
`(y-k)^2 = 4p(x-h)`

Vertex is at the origin and placed to the right

so vertex is (0,0) h=0, k=0

a reflective dish 24 in. wide and 5 in. deep.

Total 24 in wide . so 12 in on both sides (top and bottom) from vertex (0,0)

so two points on parabola is (5,12) and (5,-12)

Plug in (5,12) in the equation and find out 'p'. Also we know h=0, k=0

`(y-k)^2 = 4p(x-h)`

`(12-0)^2 = 4p(5-0)`

144= 20p

Divide both sides by 20

So P= 7.2 in

The microphone should be placed at 7.2 inches