Modeling With Periodic Functions Assignment 2

Assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate. Use the graph to complete the statements. The blades of the windmill turn on an axis that is

20 feet from the ground The blades of the windmill are 5 feet long The blade makes one revolution is 30 seconds.

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Which periodic function, sine or cosine, would be a simpler model for the situation? Explain.

A cosine function would be a simpler model for the situation.The minimum depth (low tide) occurs att = 0. A reflection of the cosine curve also has a minimum at t = 0.A sine model would require a phase shift, while a cosine model does not.

Use the graph to write an equationy = acos[b(x - h)] + k to model the situation. (temperature/month graph)

The amplitude is 30 a = -30 The period is 12 b=pi/6 [A] y = -30cos[pi/6(x +1)] +50

Write a cosine model, d = acos(bt) + k, for the depth as a function of time.

This amplitude is 1.5 meters. a=-1.5 The period is 12.5 hours. b=4pi/25 The vertical shift is k = 4 meters [D] d= -1.5cos(4pi/25t)+4

Which situations can be modeled with a periodic function?

[A] the height of a flag on one blade of a windmill [C] the height of a ball suspended from a spring

Which graph could model the path of a pendant attached to the outer edge of a wheel rim with a 7-inch radius if it takes 4 seconds for the wheel to make 1 revolution, and the tire is 4 inches thick?

[B]

Graph the equation d= -1.5cos(4pi/25t)+4 that models the situation using a graphing calculator and use it to answer the following questions. How many times during this day is the depth at the end of the pier equal to 4 meters? At approximately what time on the next day does the depth first reach 4 meters?

[C] 4 times [B] 4:00 a.m

Starting at its rightmost position, it takes 1 second for the pendulum of a grandfather clock to swing a horizontal distance of 12 inches from right to left, and 1 second for the pendulum to swing back from left to right. Write a cosine function, d = acos(bt), to model the distance, d, of the pendulum from the center (in inches) as a function of time t (in seconds).

a = 6 The period is 2 seconds b=pi t=0.5 sec d= 4.243

The blades of a windmill turn on an axis that is 40 feet from the ground. The blades are 15 feet long and complete 3 rotations every minute. Write a sine model, y = asin(bt) + k, for the height (in feet) of the end of one blade as a function of time t (in seconds). Assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate.

a is the length of the blade The vertical shift, k, is the height of the windmill a=15 k=40 The period is 20 seconds. b=pi/10 [C] y=15sin(pi/10 t) + 40

The table shows the average daily temperature on the first day of each month for one year. Use the graphing calculator to graph the data points. Use 0 for January, 1 for February, 2 for March, and so on. The graph of these points follows a path resembling a

cosine curve reflected across the x-axis