A Dish TV satellite dish is the shape of a paraboloid. The dish is 42 inches wide, and 10 inches deep. How many inches should the receiver be located from the vertex for optimal reception?

Answer 1

If I could show a graph on this editor, I would. So I'm just going to ask you to picture a parabola with its vertex at the origin, for the sake of simplicity. The symmetry of the parabola tells us that when the depth (which is actually height) is 10 inches, that the edge is 21 inches (half of 42) to the right (and to the left) of the axis of symmetry. This means that this parabola passes through (21, 10).

What we're wanting to know is how high the focus is above the vertex. The conic form of a parabola with its vertex at the origin is y = (1/(4c))x², where c is the distance between the focus and the vertex.

We can then plug in our coordinates of (21, 10), and solve for c.

10 = (1/(4c))(21²)

40c = 441

c = 441/40 = 11.025

So the receiver should be 11.025 inches above the vertex so that it's right at the focus.

One way to check this is to see if the horizontal line that passes through the focus intersects the parabola at the point (22.05, 11.025). This would verify that the focal width (also called, I kid you not, the latus rectum) is equal to 4c. This can easily be verified on a graphing calculator.