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Word Problems: Modeling with Sinusoids II
In order to solve problems which require a sinusoidal model, it is necessary to
A typical problem requiring a sinusoidal model is a relationship between time and some other data. We are given some information about data values that repeat over a certain interval or period of time.
Scientists believe that the average temperatures at various places on Earth vary from cooler to warmer over thousands of years of gradual climate change. Suppose that at one place, the highest average temperature is 80º and the lowest is 60º. Also suppose that the time it takes to go from the high to low average is 20,000 years, and in the year 2000 the average temperature is at a high point of 80º. How can we use a sinusoidal expression to model this phenomenon?
Let T(t) be the temperature in a given year t. We can model this situation using
  • A is the amplitude
  • f is the frequency which is
  • t is the year
  • C/2pf is the phase shift
  • D is a vertical displacement (from the temperature of 0º).
Sometimes this is written as
  • B = 2pf
  • C/B is the phase shift, which is 2000 years to the high point of the cosine function.
We first notice that the range of temperatures is 20º. D = 70° is the average temperature around which the highs and lows vary. The temperature varies 10º above and below this mid-point. The variation is given by A = 10º which is the amplitude or amount of temperature change above and below 70º.
Second, the period of this temperature change is 40,000 years because that is the time it takes to go from 80º, then down to 60º, and finally back to (repeating) 80º. Therefore
We now have
We find C by noting that the value of this expression must be 80 when t = 2000. This means that
must equal 1 when t = 2000. This happens when
We could also set C/B = 2000 and solve for C, remembering that we have
Our final equation modeling this phenomenon is
This is a bit ugly, but it works! When we let t = 2000 we get
which was given as the average temperature in the year 2000.
We could also rewrite this expression by factoring out the
which more clearly shows that at t = 2000, T(t) = 80.
A graph is shown below using -80,000 < x < +80,000 counting by units of 20,000. A horizontal line is drawn at 70 indicating the midpoint of the average high and low temperatures. This calculator graph requires that the MODE be set for Radians. Notice that for year 2000 the average temperature is 80.

Let's Practice

Question #1
A city averages 14 hours of daylight in June, 10 in December, and 12 in both March and September. Assume that the number of hours of daylight varies sinusoidally over a period of one year. Write an expression for n, the number of hours of daylight, as a cosine function of t. Let t be in months and t = 0 correspond to the month of January.

Question #2
The average depth of water at the end of a dock is 6 feet. This varies 2 feet in both directions with the tide. Suppose there is a high tide at 4 AM. If the tide goes from low to high every 6 hours, write a cosine function d(t) describing the depth of the water as a function of time with t = 4 corresponding to 4 AM. (Note: this is an approximation since the actual time between high and low tides is not exactly 6 hours).

Try These
Question #1
Astronomers have noticed that the number of visible sunspots varies from a minimum of about 10 to a maximum of about 110 per year. Further, this variation is sinusoidal, repeating over an 11 year period. If the last maximum occurred in 2003, write a cosine function n(t) which models this phenomenon in terms of the time t which represents the year.

Question #2
AudioAn electric current alternates with a frequency of 60 cycles per second. This is called quite aptly alternating current and is the type of electrical system we have in our homes and offices in the United States. Suppose that at time t = 0.01 seconds, the current is at its maximum of I = 5 amperes. If the current varies sinusoidally over time, write an expression for I amperes as a function of t in seconds.

This type of problem requires an understanding of the parts of the basic function model given by either where A is the amplitude and D is the midpoint between high and low points about which the values of y oscillate. There is a periodic change in y-values, meaning that the entire range of y-values repeats in a given period of time. This time interval is called the period of the function. The frequency f of this function is the reciprocal of the period and B = 2pf. The phase shift is the t location of a maximum point (if using cosine) and is the same as C/B. If you were to use a minimum point, we would place a negative sign in front of A when writing the final expression.
Note that it is often easier to use cosine rather than sine to model this type of periodic behavior. Use of a sine function requires use of a midpoint value rather than a maximum or minimum value for the function. We have not included examples which use a sine function as a model.

M Ransom

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