Worksheet #2. Use trigonometry and logarithms to model natural phenomena from a periodic behavior perspective.

Transcription

1 math 112 Worksheet #2 Name Due Date: Objective: Use trigonometry and logarithms to model natural phenomena from a periodic behavior perspective. We revisit the varying of tree ring widths over time. We update our model by dealing with some gradual change over time. First we must find the trigonometric function that matches the following data, just like the previous worksheet. Consider a tree with tree ring widths that vary in a periodic manner. Suppose a core sample from a living tree gives a tree ring pattern that generates the following table of data. Table 1 : tree ring data Let t be the number of years after the year 2000 and y be the width of the tree rings in millimeters years after width in 2000, t millimeters, y On your calculator make a scatter plot of the data points for the tree rings. 1

2 Assume the repetitive pattern continues consistently through time. To find a periodic function that matches the data answer the following questions. Determine the midline of the scatter plot. y = Determine the amplitude for the periodic function. A = Determine the period of the periodic function. T = Create a periodic function with the above midline, amplitude and period. Ignore the phase shift for a moment. Use either sine or cosine. g(t) = 2

3 Determine an appropriate horizontal shift for matching the data points. This is where a basic cycle starts on the graph. It will be a integer number of years. C ω = State the value of the phase shift, C, for the function. Write out the periodic function that matches the data. f(t) = On your calculator graph the data points and the matching function over the interval 5 t 15. Do they match up? 3

4 Changing Averages For long lived trees the successive tree ring width reflects a decrease that is not related to environmentally sensitive factors. The average width of the tree rings displays an exponential decay with the passing of time. The following graph shows what the decrease in average width looks like. y t Figure 1: Average width versus time Assume a reasonable exponential function for the decrease in tree ring width fits the following table. Let t be the number of years after year t average width(mm) Table 2 : average tree ring widths in millimeters

5 State the exponential function for the average tree ring width. Use the exponential format y = A 0 e kt. (Find the decay constant to 5 decimal places.) When approximately will the average width be 2 mm? 5

6 We now develop a modeling function that takes the changing average into account. The exponential function found above will serve to give midline-like values. This is the form of the new function. p(t) = A q(ωt C) + Be kt, where q(t) = sin(t) or q(t) = cos(t). Assume the amplitude stays constant, which is a dangerous assumption. State the function that models the tree rings with this decreasing width. Use your earlier values for A, ω, and C. p(t) = Graph this function on your calculator. 6

7 State the closed interval of t-values where the exponential function is between 1 and 1.5 inclusive. Remember the exponential function models average width. In the interval you found above, use your grapher to find all the t-values where p(t) =

8 Without using your grapher, state an interval where you expect our model must lose its validity, namely negative width. Make this interval as small as possible. Use your grapher to find the first t-value where the p(t) is zero. 8