Word Lesson: Modeling with Sinusoids 2

In order to solve problems which require a sinusoidal model, it is necessary to

use basic graphing skills for sine and cosine
know how to find amplitude, period, frequency, displacement, and phase shift
know that use of a calculator requires that the MODE be set for Radians.

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

where

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

where

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.

Examples

	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.

	What is your answer?

	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).

	What is your answer?

Examples

	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.

	What is your answer?

	An 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.

	What is your answer?