A boat tied up at a dock bobs up and down with the passing waves. Vertical distance between its high and low points is 1.8 m. The cycle is repeated every 4 s. At what time or times during a cycle is the instantaneous rate of change of the vertical position of the boat equal to 0, and at what time is it a maximum?
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We are given that a boat bobs up and down with the waves such that the distance from highest to lowest point is 1.8 meters and the cycle is repeated every 4 seconds. We are asked to find the time(s) that the instantaneous rate of change of the vertical displacement is zero and when it achieves its maximum.
The underlying model is a sinusoid. Since there is no indication as to where or when to start, we can use a sine wave such that at t=0 the boat is midway between its highest and lowest point. We can also place the midline at y=0 as sea level.
The amplitude is .9 meters. (The boat goes up .9 meters from the midline and down .9 meters from the midline for a distance between maximum and minimum of 1.8 meters.) The period is 4 seconds (the time to repeat the action).
The function model is `y=.9sin(pi/2 t)` with `y=a*sin(bt)` where a is the amplitude and b is 2pi divided by the period. (See image for the graph.)
To determine the time when the instantaneous rate of change is zero we take the...
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