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Answer :
Final answer:
The question deals with finding times when two sound waves have the same frequency within a given interval. The times when frequencies of given waves y=2sin2x and y=ksinx align are when x = 0, 180 due to the periodic nature of the sine function and the interval [0, 360). The Correct Answer is Option.B.
Explanation:
The student's question is related to finding the times at which two sinusoidal sound waves have the same frequency within a given interval. In the provided functions y=2sin2x and y=ksinx, the frequency of the wave is determined by the coefficient of x in the sine function.
For the first wave, the frequency is doubled (due to the 2 multiplied to x), and for the second wave, the frequency is as is indicated by ksinx. Frequencies will be the same when the argument of the sine function is the same or differs by a multiple of 2π radians since the sine function is periodic with period 2π.
To find the times when the frequencies are the same, we need the arguments of the sine functions to match; this happens when 2x is equivalent to x, modulo 2π. Since we are only concerned with the interval [0, 360), and we know the sine function repeats every 360 degrees, we're looking for multiples of 360 degrees where the doubled frequency and the single frequency sine waves align.
The correct times would be when x = 0, 180 because these are the points within the given interval where sin(2x) = sin(x) and x = 0, 180, 360, ... where the latter exceeds the interval. If x = 180, then 2×180 = 360 which is equivalent to x = 0 due to periodicity. Hence, the correct answer would be b) x=0, 180, 210, 330.
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