Question

A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the exact width (in inches) of the dish if the depth is to be 2 feet.

Answer 1

First, lets remember that a paraboloid is just a parabola that was rotated around the Y axis. So lets find the equation from our parabola that is been rotated. We know that the focus is in the point (0,5), so we can find our equation:

$y=\frac{x^2}{4\times5}\rightarrow y=\frac{x^2}{20}$

Just to remember, a parabola that has focus in (0,P) has equation as:

$y=\frac{x^2}{4\times p}$

We can draw our parabola:

So we want to find the width knowing that the depth is 2 feet our 24 inches. That means that he wants to know the X that make the Y be 24, so lets find it:

$y=\frac{x^2}{20}\rightarrow x^2=24\times20\rightarrow x=\pm\sqrt[]{480}\rightarrow x^{\prime}\cong21.908\text{ and }x\~\~^{}\cong-21.908$