677 CHAPTER 6 Review Exercises 0 0 600 6 107. (Modeling) Pollution Trends The amount of pollution in the air is lower after heavy spring rains and higher after periods of little rain. In addition to this seasonal fluctuation, the long-term trend is upward. An idealized graph of this situation is shown in the figure. 108. (Modeling) Lynx and Hare Populations The figure shows the populations of lynx and hares in Canada for the years 1847 – 1903. The hares are food for the lynx. An increase in hare population causes an increase in lynx population some time later. The increasing lynx population then causes a decline in hare population. The two graphs have the same period. Year Canadian Lynx and Hare Populations Number 150,000 100,000 50,000 1850 1860 1870 1880 1890 1900 Hare Lynx (a) Estimate the length of one period. (b) Estimate the maximum and minimum hare populations. An object in simple harmonic motion has position function s1t2, in inches, from an equilibrium point, where t is time in seconds. Find the amplitude, period, and frequency. 109. s1t2 = 4 sin pt 110. s1t2 = 3 cos 2t 111. In Exercise 109, what does the frequency represent? Find the position of the object relative to the equilibrium point at 1.5 sec, 2 sec, and 3.25 sec. 112. In Exercise 110, what does the period represent? What does the amplitude represent? Circular functions can be used to model the fluctuating part of the pollution levels. Powers of the number e (e is the base of the natural logarithm; e ≈2.718282) can be used to model long-term growth. The pollution level in a certain area might be given by the formula y = 711 - cos 2px21x + 102 + 100e0.2x, where x is time in years, with x = 0 representing January 1 of the base year. July 1 of the same year would be represented by x = 0.5, October 1 of the following year would be represented by x = 1.75, and so on. Find the pollution level on each date. (a) January 1, base year (See the figure.) (b) July 1, base year (c) January 1, following year (d) July 1, following year
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