Tarzan Problem:Tarzan is swinging back and forth on his grapevine. As he swings,

Question

Tarzan Problem:Tarzan is swinging back and forth on his grapevine. As he swings, he goes back and forth across the river bank, going alternately over land and over water. Jane decides to model his motion mathematically and starts her stopwatch. Let t be the number of seconds the stopwatch reads. Let y be the number of meters Tarzan is from the river bank, with y being positive when he is over the water and negative when he is over land. Assume that y varies sinusoidally with t. At t=2, Jane finds that Tarzan is at one end of his swing, 23 feet over the land. She finds that when t=5, he reaches the other end of the swing 17 feet over the water. Sketch the graph of the sinusoid labeling important points on the axes, state the amplitude, period, phase shift, and vertical shift of the sinusoidal function. Finally, write an equation for the distance from the river bank as a function of time.
Tarzan Problem:Tarzan is swinging back and forth on his grapevine. As he swings, he goes back and forth across the river bank, going alternately over land and over water. Jane decides to model his motion mathematically and starts her stopwatch. Let t be the number of seconds the stopwatch reads. Let y be the number of meters Tarzan is from the river bank, with y being positive when he is over the water and negative when he is over land. Assume that y varies sinusoidally with t. At t=2, Jane finds that Tarzan is at one end of his swing, 23 feet over the land. She finds that when t=5, he reaches the other end of the swing 17 feet over the water. Sketch the graph of the sinusoid labeling important points on the axes, state the amplitude, period, phase shift, and vertical shift of the sinusoidal function. Finally, write an equation for the distance from the river bank as a function of time.

Explanation / Answer

Recall that a generalized cosine equation is of the form:

y = acos[b(x - c)] + d, where a describes the amplitude, b helps determine the period, c determines the horizontal shift, and d describes the vertical shift.

1) To summarize your data, we have two points of the form (t, y(t)) that are: (2, -23) and (5, 17). Amplitude is half the distance traveled from one end of his swing to the other.

Thus, a = (17 - (-23))/2 = 40/2 = 20. BUT, since he starts on the negative (y = -23), then this implies a must be negative (because cosine normally starts positive). So a = -20.

Recall that the period is the time needed to complete one full cycle and that period = 2Pi/b. Since he travels from one end to the other in 5 - 2 = 3 seconds, it would take him 6 seconds to travel back to where he started.

Thus, 6 = 2Pi/b ---> b = 2Pi/6 = Pi/3.

For the shifts, horizontally, we know it starts at t = 2, so, this is your c. Vertically, since we know the amplitude is 20 (and this makes the lowest point y = -20 normally), and the lowest point is y = -23 (differing by 3), then the vertical shift is down 3. Thus d = -3. Putting it all together,

y = -20cos[Pi/3(x - 2)] - 3, or y = -20cos(Pi*x/3 - 2Pi/3) - 3

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