Periodic behavior is common in nature. For example, animal populations, sound waves, and the tides all exhibit periodic behavior. The ocean flows from high tide to low tide, then back over and over again. This motion can be modeled by trigonometric functions. Follow the directions below to explore one such example.
Throughout the day the depth of water at the end of a pier varies with the tides. High tide occurs at 4:00 a.m. with a depth of 6 meters. Low tide occurs at 10:00 a.m. with a depth of 2 meters.
1. Model the problem by using the given trigonometric equation to show the depth (y) of the water x hours after midnight, showing all your work. y = A cos(Bx + C) + D
Start by sketching a graph of the situation – sketch 2 cycles. (Pick appropriate intervals for the x- and y-axes and make the horizontal axis in time, not radians. Hint: What time should x = 0 be?)
Use the above graph and any extra work needed to determine the amplitude, period, and horizontal shift, and vertical shift to model the equation. Period and phase shift must be in radians.
Amplitude = _________
Period (in time) = ________ convert period to radians: ___________________________
Horizontal shift (in time) = ________ convert phase shift to radians: _______________________
(To find the phase shift use: -CB=x, where x is the horizontal shift in time.)
Vertical shift = _________
Equation: __________________________________________________________
2. A large boat coming in at noon needs at least 4 meters of water to dock at the end of the pier. Will the boat be able to safely dock? Solve the problem by using the equation to find the exact depth of the water at noon. Explain your reasoning.
Show work below: (Hint: how much time after x=0 is noon?)
Will the boat be able to dock safely? _______________________________________________________
Explain your answer/reasoning: ___________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
3. Color a fun dock/pier ocean-scape on your graph.
Answer:
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