Satellite dish is shaped like a paraboloid
of revolution. The signals are emanate From a satellite strike the surface of the dish and our reflected to a single point, where the retriever is located. The dish is 12 feet across and it’s opening and 4 feet deep at its center, what position should the retriever be placed?

To find where the retriever is located, you need to find the focus point of the parabola. Our parabola equation is equal to 1 = [tex]\frac{x^{2} }{a^{2}}[/tex] + [tex]\frac{y^{2} }{b^{2}}[/tex] where a and b are constants that shape the parabolid. Something to note about paraboloids is that when a ray hits a satellite dish, it will always reflect to the focus point in the center of the satellite dish because of it's reflexive properties so that's why we're finding the focus.

The equation for the distance p of the focus point (or directorix) from the vertex is [tex]\frac{1}{4a}[/tex] or [tex]\frac{1}{4b}[/tex] depending on the orientation of the paraboloid, vertical or horizontal. If the paraboloid opens along the y-axis, (horizontal) you'd use [tex]\frac{1}{4b}[/tex] and vice versa for the vertical axis. This also can depend how you orient it, and I'll put a drawing below to help visualize it.