Makena Preston

2022-07-17

Parabolic reflectors with microphon located at the focus are used to capture sounds from a distance. The sound waves enter the reflector and are concentrated toward the microphone. How far from the reflector should a microphone be placed if the reflector has a width of 3 feet and a depth of 1.25 feet?

hottchevymanzm

Expert

The parabola is horizontal so the standard form is:
$\left(y-k{\right)}^{2}=4p\left(x-h\right)$
Since the vertex is at (h, k) and the focus is at (h +p, k), then the microphone must be placed |p feet to the right of the vertex. If the vertex is at (0,0), then h = 0 and k = 0:
${y}^{2}=4px$
To find p, substitute a point. A point on the parabola is (1.25, 1.5) since the depth is 1.25 ft and the width is 3 m (half of it).
Hence,
$\left(1.5{\right)}^{2}=4p\left(1.25\right)$
2.25=5p
0.45=p
So, the microphone should be placed 0.45 ft to the right of the vertex of the reflector.
Result:
0.45 ft to the right of the vertex of the reflector

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