parabola equation

Question

An observatory has a large antenna to collect signals from space. The antenna has a curved surface with a parabolic cross section, and a microphone is located at the focus of the parabola. The microphone is 10.25 feet from the vertex of the parabola. Write an equation that can be used to model the parabola. Explain how you found your equation.

Answer 1

Hello Derek,

Suppose we locate a parabola in a rectangular coordinate system. If the focus of the parabola is the point (0,p), and the directrix of the parabola is the horizontal line y = -p, then its equation may be written as

x² = 4py,

or, equivalently,

y = (1/(4p))x².

Note also that the vertex of this parabola is at (0,0), and the axis of symmetry is the y-axis. So for the parabolic cross section of the antenna described in your problem, set up a coordinate system so that the vertex is at the origin (0,0), the axis of symmetry is the y-axis, and the focus is (0,10.25). Thus, p = 10.25, and using the equation above, we have

y = (1/(4·10.25))x²

y = (1/41)x²

Hope that helps! Let me know if you need any further explanation.

William