Answer to Question #255562 in Analytic Geometry for Joshua

Question

1. Mrs. Flores recently subscribed in a tv cable plan at GSAT Company. The receiving dish of the GSAT Cable is in the shape of a paraboloid of revolution. Find the location of the receiver which is placed at the focus if the dish is 12 feet across and 3 feet deep.

Answer 1

If a satellite dish is 12 feet across and 3ft deep, how far from the bottom of the dish should the receiver be placed so that it is at the focus of the paraboloid?

First let's consider what this looks like in two dimensions. Because we are setting up the graph of this parabola, we can choose to place the vertex at the origin. This makes things a little easier.

since the dish was a total of 12 feet across, we split this between the two sides of the graph, creating the points (6,3) and (−6,3). since the vertex for this parabola is at the origin, the standard equation is somewhat simplified.

Taking the standard equation for an upward facing parabola: 6p(y−k)=(x−h)² and using the point (0,0) for the vertex leaves us with 6p(y−0)=(x−0)² or 6py=x²

The main thing we need to do is find the value of p for this situation. This will tell us where the receiver should be on the satellite dish. Plugging in the points on the graph that we know will allow us to solve for p.

$\begin{aligned} 6 p y &=x^{2} \\ 6p(3) &=(6)^{2} \\ 18p &=36 \\ p &=2 \end{aligned}$

Thus the receiver should be 2 foot from the bottom of the dish$.$