Sound Broadcasters use a parabolic microphone on football sidelines to pick up field audio for broadcasting purposes. A certain parabolic microphone has a reflector dish with a diameter of 28 inches and a depth of 14 inches. If the receiver of the microphone is located at the focus of the reflector dish, how far from the vertex should the receiver be positioned?
1. What is the diameter of the reflector dish?
2. What is the depth of the reflector dish?
3. What is the problem asking you to determine?
4. Sketch a graph of a vertical parabola to represent the reflector dish. Place the vertex at the origin.
5. You know the coordinates of two other points on the parabola. Plot and label them on your graph.
6. What is the equation for a vertical parabola with vertex at the origin?
7. How can you find the location of the focus from the equation for the parabola?
8. What is the location of the focus?
9. If the receiver of the microphone is located at the focus of the reflector dish, how far from the vertex should the receiver be positioned?
1. The diameter of the reflector dish is 28 inches.
2. The depth of the reflector dish is 14 inches.
3. The problem is asking to determine how far from the vertex should the receiver be positioned if it is located at the focus of the reflector dish.
4. (See attached image for graph)
5. Two other points on the parabola are (14, 14) and (-14, 14).
6. The equation for a vertical parabola with vertex at the origin is y = ax^2.
7. The equation for a vertical parabola with vertex at the origin can be written in the form x^2 = 4py, where p represents the distance from the vertex to the focus.
8. From the equation x^2 = 4py, we can see that p = 7 inches, so the focus is located at (0, 7).
9. If the receiver of the microphone is located at the focus of the reflector dish, it should be positioned 7 inches from the vertex.