Parabolic Microphone

Congratulations, you were able to ride your motor bike to a forested area just outside the V.A.L.A. camp without detection. Now you must spy on them to learn their plans for the Fire Nosed Bear. You brought along your folding parabolic microphone but you lost the instructions for how to assemble it!

A parabolic microphone works by bouncing the sound waves off of the dish toward the microphone in the center. A parabola curve has the unique property that all waves coming straight in get bounced toward the central focal point (point F below) no matter where they hit the dish! But you need to figure out where that central point is--how far above the base must you place the microphone?

Your parabolic microphone is 50 cm wide and about 15 cm deep. The equation that describes the dish from the side is y = x²/40. First graph your parabola on graph paper with (x=0,y=0) at the bottom center of your paper.

The steepness of the curve is called the slope--like a ski slope--where the bigger the number, the steeper the slope. In math, you can get an equation that gives you the slope at every point x if you have the equation of that curve--which you do! This is called the derivative. Taking the derivative of your parabola equation gives you this:  slope = x/20. Now calculate the slope at x={0,5,10,15,20,24} and add those values to your graph.

To figure out where to place the microphone, you only need to figure out where one of the sound waves would bounce off the dish and end up. The easiest point is where the wave comes down from the top (in the diagram above) and bounces off the dish at a 45 degree angle and goes horizontally to the microphone. So you just need to find the point on the curve where the slope is 45 degrees. Unfortunately, the slope from the derivative (equation above) gives the slope as a ratio of the rise over the run, as shown in the diagram to the right. What is the slope that corresponds to a 45 degree angle? Once you find that, you can find the point that is at the same height as the microphone. Then you can finish building your parabolic microphone and listen in on the V.A.L.A. agents!