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WAVES: SINE CURVE APPLICATIONS
Real - World Applications
:Employment Cycles at Securities Firms and Temporary Hiring
Modeling the real world: The sine curve function is representative of a class of functions that can be used to model cyclical or periodic behavior. Applications of this type of function are numerous and varied. These applications range from fluid flow and wave motion to sound waves, tides, AC currents, and medical utilities such as the output from a heart patient's EKG machine. It is interesting to note that the various applications of periodic functions also extend to economics and business applications.
In the following two examples we present economics applications of periodic functions. The first example is a histogram of data that appears to follow a periodic function. The second example presents a specific sine function denoting applications for temporary employment. Please note the practical interpretations of the periodic parameters in the second example.
(These examples have been inspired from the online web page of Stefan Wagner and Steven R. Costenoble, Hofstra University, http://www.hofstra.edu/~matscw/trig/trig1.html.)
- Cyclic Commodity Prices:
The histogram below plots U.S. city average ground chuck, 100% beef, prices (dollars per pound) in January versus the years from 1980 to 1999. The data is from the Bureau of Labor Statistics. One can observe that the heights of these rectangles trace out a curve similar to the sine curve from the applet.
- Cyclical Employment Patterns for Temporary Hiring Firms:
Suppose an economist consulted by a temporary employment agency indicates that the demand for temporary employment (measured in thousands of job applications per week) can be modeled by the function defined by
y = 4.3sin(0.82t + 0.3) + 7.3,
where t is time in years since January 1998. We note that this function is a generalized sine curve and thus we can determine the amplitude, the vertical offset, the phase shift, the angular frequency, and the period with the following results:
From our generalized form: y = a sin[b(t-c)] + d, and our specific function
y = 4.3sin(0.82t+0.3) + 7.3
We note that
y = a sin[b(t-c)]+d = a sin[bt-bc] + d,
Thus, amplitude a = 4.3, vertical offset d = 7.3, and angular frequency b = 0.82.
Note also that bc = 0.3, which implies that
phase shift c = 0.3/b = 0.3/0.82 @ 0.37.
To determine the period we use the formula b = 2p / p.
So 0.82 = 2p /p which implies
period p = 2p /0.82 @ 7.7.
Of most interest is the practical interpretation of these numbers.
What do these constants mean?
We can combine the values of the constants above to give a short description of the behavior of this employment model. For example, we may conclude that the demand for temporary employment cycles every 7.7 years with an "average" or baseline of 7,300 job applications per week.
We can also note that during each cycle the demand reaches a peak of 11,600 applications (4,300 applications above the baseline) and dips to a low of 3,000. One can observe that in April, 1998 (t=0.37) the demand for employment was at the average of baseline level and on an upward trend.
USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil