Mathematics

Mathematics

Problem

75. Parabolic Microphones B-Ball Network uses a parabolic microphone to capture all the sounds from the basketball players and coaches during each regular season game. If one of its microphones has a parabolic surface generated by the parabola 18y=x218 y=x^{2}, locate the focus (the electronic receiver) of the parabola.
Intuitive explanation

A parabola's focus is the special point where all sound waves reflecting off the curved surface converge—that's why it's the ideal spot for the receiver. To find it, you need to recognize the standard form of your parabola and extract the focal parameter. Rewrite 18y=x218y = x^2 as x2=18yx^2 = 18y, then compare it to x2=4pyx^2 = 4py to identify pp, which tells you exactly how far the focus sits above the vertex.

Intuitive explanation

A parabola's focus is the special point where all sound waves reflecting off the curved surface converge—that's why it's the ideal spot for the receiver. To find it, you need to recognize the standard form of your parabola and extract the focal parameter. Rewrite 18y=x218y = x^2 as x2=18yx^2 = 18y, then compare it to x2=4pyx^2 = 4py to identify pp, which tells you exactly how far the focus sits above the vertex.

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