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Precalculus HS Mathematics Unit: 04 Lesson: 02

Applications of Sinusoidal Functions (pp. 1 of 4)

1) To power a combustion engine, fuel creates a tiny explosion that pushes a piston down a cylinder. The crank shaft is turned, which pushes the piston back up, and the cycle starts over again, repeating many times every second. Suppose the up and down motion of a piston in a running engine can be modeled by the function d (t ) 3.8 sin(50t ) , where t is time in seconds, and d(t) gives the displacement (in cm) of the piston above or below its level at rest. A) What is the amplitude of this sinusoidal function? Interpret this in the context of the problem.

+

A piston in the cylinder

4 pistons on a crank shaft

B)

What is the period of this sinusoidal function? Interpret this in the context of the problem.

C)

If the piston completes an up-and-down cycle at this rate (or, over this period), how many revolutions would the crank shaft turn per minute (rpm)?

D)

Determine a good graphing window to view several cycles of this function.

E)

Many engines run on four cylinders. This means that 4 pistons are attached to the crank shaft, and must be engineered where they rise and fall at even time intervals so that the engine runs "smooth." (See the figure above, and graph below.) Find three other sinusoidal functions that could be used to model the displacement of the other pistons in this engine (if the first piston uses the original function). Y1 3.8 sin(50t ) Y2 = _________________________ Y3 = _________________________ Y4 = _________________________

©2010, TESCCC

08/01/10

Precalculus HS Mathematics Unit: 04 Lesson: 02

Applications of Sinusoidal Functions (pp. 2 of 4)

2) At an amusement park, the Ferris wheel has a diameter of 42 ft, and it is positioned 5 feet above the ground. To ride costs 7 tickets, so Charlie wants to time the ride to make sure he gets his money's worth. Once all the passengers are loaded, the ride begins, and Charlie starts his stopwatch. He reaches the top 9 seconds later, and returns to the top when the stopwatch reads 33 seconds, and again at 57 seconds. While on the Ferris wheel, Charlie's height above the ground (h, in feet) is a sinusoidal function of time (t, in seconds). A) Using the information in the description above, sketch the graph of this function on a time interval of [0, 100]

42 ft

5 ft

B)

What is the period of this function? Why?

C)

Write and function of the form h(t ) a cos(b(t c )) d to model this situation.

D)

What was Charlie's height above the ground when the ride started?

E)

The ride lasted 2 minutes and 5 seconds before it stopped to begin letting passengers off the Ferris wheel. At this point, Charlie dropped his stopwatch over the edge of his seat on the Ferris wheel. From what height did the stopwatch fall?

©2010, TESCCC

08/01/10

Precalculus HS Mathematics Unit: 04 Lesson: 02

Applications of Sinusoidal Functions (pp. 3 of 4)

3) A spring is used to suspend a weight from above. As a physics experiment, a student drops the weight, then begins recording its movement with a motion detector. She notices that the weight bounces back and forth between heights of 25 cm and 7 cm, at specific intervals of time, as shown in the table. t (sec) h(t) (cm) 0 11.5 0.5 7 1 11.5 1.5 20.5 2 25 2.5 20.5 3 11.5 3.5 7

h = 25 cm h = 7 cm

Assume that the height of the weight is a sinusoidal function of time in seconds. A) Sketch a graph of this function on the interval 0 t 12.

B)

Write and function of the form h(t ) a cos(b(t c )) d to model this situation.

C)

At what times was the weight at a height of 16 cm?

D)

After the experiment began, what were the first two values of time at which the weight was exactly 20 cm high? For what length of time in each cycle does the weight stay at or above 20 cm in height?

©2010, TESCCC

08/01/10

Precalculus HS Mathematics Unit: 04 Lesson: 02

Applications of Sinusoidal Functions (pp. 4 of 4)

4) While vacationing in Port Aransas, Shelly noticed that the water level relative to the pier changed throughout the day. She recorded the following information about high tide and low tide over a two-day period. A)

Water Level Chart Water Day Time Level Mon. 5:42 AM -2.4 ft Mon. 6:12 PM -2.1 ft Tues. 6:42 AM -2.4 ft Tues. 7:12 PM -2.1 ft

Low Tide: h = -2.4 ft High Tide h = -2.1 ft

Let t = time in hours since midnight on Monday. This means that 5:42 AM Monday would correspond to t = 5.7. What values of t could be used to represent the other times on the chart above? What is the period of time over which the tides cycle? Write a sinusoidal function that relates the water level to t. Use the function to predict the water level at 8:00 AM on Wednesday.

B) C)

5)

One fine summer Saturday in the Texas panhandle, the outdoor temperatures ranged from 65F to 93F, according to the table. Assume that, on this day, temperature is a sinusoidal function of time. A) Let x = time in hours since midnight on Saturday. Find a function that gives the temperature in terms of x.

Saturday Temperatures Time 5:00 AM Temp. 65F

LOW

7:00 AM 9:00 AM 11:00 AM 1:00 PM 3:00 PM 5:00 PM

67F 72F 79F 86F 91F 93F HIGH

B)

At what time on Saturday night did the temperature drop back down to 80F?

C)

Use the function model to predict the temperature at 4:00 AM on Sunday morning.

©2010, TESCCC

08/01/10

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