Periodic waves spread out over the surface of a lake where two women, Jenny and Melissa, are fishing in separate boats 102 meters apart. Each woman's boat bobs up and down 19/min. At a time when Jenny's boat is at a crest, Melissa's boat is at its lowest point, and there are 4 additional crests between them. Calculate the wavelength of these water waves.
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To calculate the wavelength of the water waves, we need to determine the distance between two adjacent crests.
We know that there are 4 crests between Jenny and Melissa's boats, and the distance between the boats is 102 meters. We also know that the boats bob up and down 19 times per minute.
First, let's calculate the distance traveled by the waves between two adjacent crests. Since one complete wave cycle consists of a crest and a trough, we can divide the distance between the boats by the number of wave cycles.
Distance between adjacent crests = Distance between the boats / Number of wave cycles
Distance between adjacent crests = 102 meters / 4 wave cycles
Distance between adjacent crests = 25.5 meters
Next, let's calculate the period of the wave, which is the time it takes for one complete wave cycle. Since the boats bob up and down 19 times per minute, the period of the wave can be calculated as 1/19 or approximately 0.0526 minutes (rounded to four decimal places).
Now, we can calculate the speed of the wave using the formula:
Speed = Wavelength x Frequency
The frequency is the number of wave cycles per unit of time, which in this case is the number of wave cycles per minute. We know that there are 19 wave cycles per minute.
Speed = 25.5 meters x 19 wave cycles/minute
Speed = 484.5 meters/minute
Finally, we can rearrange the formula to solve for the wavelength:
Wavelength = Speed / Frequency
Wavelength = 484.5 meters/minute / 19 wave cycles/minute
Wavelength ≈ 25.5 meters
Therefore, the wavelength of these water waves is approximately 25.5 meters.