Word Problems: Modeling with Sinusoids I 
In order to solve problems which require a sinusoidal model, it is necessary to A typical problem requiring a sinusoidal model in which period, frequency and amplitude are important is a relationship between time and some other data. In many situations, this involves motion which repeats or oscillates. We are given some information about data values that repeat over a certain interval or period of time, or we are given information about the position of an object that varies sinusoidally. Suppose a particle moves along the xaxis. Its position (xcoordinate) at any time t seconds where t is greater than or equal to zero is given by . (a) What is the position of the particle at time t = 2.3 seconds? (b) What are the amplitude, period and frequency of this motion? (c) What is the smallest value of x that the particle reaches during its motion?
(a) To find the position of the particle at t = 2.3 we evaluate
Note that this tell us the xcoordinate of the particle at t = 2.3.
.
.
From this we see that
Since we can determine that the period is
T = 2 seconds.
(c) The maximum distance this particle moves can be seen easily if we note that at time t = 0 the particle is at the coordinate x = 0. We call this the stable (equilibrium) position of the particle since it moves to the left and right of this position, which acts as a center of the motion. Since the amplitude of the motion is 2, the particle moves from the origin at most a distance of 2. This means the smallest value of x that the particle reaches is x = 2. The particle moves back and forth between the xcoordinates 2 and +2 in a period of 2 seconds. A graph of the position of this particle is shown below over a 10 second time interval.
Remember that the calculator uses X instead of t. So the expression Y1=2sin( pX) really represents Y1=2sin( pt) or the values for our function s(t). That is, the values of Y1 are the xcoordinates of the particle's position as it moves along the xaxis. Notice that at time t = 2.3 the particle's approximate position, or xcoordinate, is 1.618.

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This type of problem requires a thorough knowledge of the equation s(t) = Asin(2pft). It is often easier to obtain the value of A, the amplitude, first. The value of f can be found by determining the period, or time interval required for the motion of the object to travel once through all possible positions. It is also important to remember that f = 1/period, that is, the frequency is the reciprocal of the period. Notice that depending upon which information we know, these problems can require as few as two steps, or as many as four steps to solve.

M Ransom


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