Honors PrecalculusName:
Applications of Sine and Cosine Graphs
1. For a person at rest, the velocity v (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by
v = 0.85 sin , where t is the time (in seconds). (Inhalation occurs when v > 0, and exhalation occurs when v < 0.)
a. Find the time for one full respiratory cycle.
b. Find the number of cycles per minute.
c. The model is for a person at rest. How might the model change for a person who is exercising? Explain.
2. A company that produces snowboards, which are seasonal products, forecasts monthly sales for 1 year to be S = 74.50 + 43.75 cos , where S is the sales in thousands of units and t is the time in months, with t = 1 corresponding to January.
a. Graph on your calculator the sales function over the one-year period.
How is your window set?x min:x max:
y min:y max:
b. Determine the months of maximum and minimum sales.
3. The daily consumption C (in gallons) of diesel fuel on a farm is modeled by
C = 30.3 + 21.6 sin ( + 10.9) where t is the time in days, with t = 1 corresponding to January 1.
a. Without finding the period, what do you think it should be?
b. What is the period of the model? Is it what you expected? Explain.
c. What is the average daily fuel consumption? Which term of the model did you use? Explain.
d. Graph the model on your calculator. Use the graph to approximate the time of year when
consumption exceeds 40 gallons per day.
4. The percent y (in decimal form) of the moon’s face that is illuminated on
day x of the year 2006, where x = 1 represents January 1, is shown in the table. (Source: U.S. Naval Observatory)
Day, x / Percent, y29 / 0.0
36 / 0.5
44 / 1.0
52 / 0.5
59 / 0.0
66 / 0.5
a. Create a scatter plot of the data on your calculator.
b. Find a trig model for the data (without using the calculator regression!).
c. Graph your equation from part b to the scatter plot. How well does the model fit the data?
d. What is the period of the model?
5. The table shows the average daily high temperatures for Nantucket, Massachusetts N and Athens, Georgia A (in degrees Fahrenheit) for month t, with t = 1 corresponding to January. (Source: U.S. Weather Bureau and the National Weather Service)
Month, t / Nantucket, N / Athens, A1 / 40 / 52
2 / 41 / 56
3 / 42 / 65
4 / 53 / 73
5 / 62 / 81
6 / 71 / 87
7 / 78 / 90
8 / 76 / 88
9 / 70 / 83
10 / 59 / 74
11 / 48 / 64
12 / 40 / 55
a. A model for the temperature in Nantucket is given by N(t) = 58 + 19 sin . Create a scatter plot on your calculator and find a model for Athens (without using the calculator regression!).
b. Graph the data and the model for the temperatures in Athens in the same viewing window. How well does the model fit the data? What could be improved?
c. Use the models to estimate the average daily high temperature in each city. Which term of the models did you use? Explain.
d. What is the period of each model? Are the periods what you expected? Explain.
Nantucket:Athens:
e. Which city has the greater variability in temperature throughout the year? Which factor of the models determines the variability? Explain.