487 4.4 Evaluating Logarithms and the Change-of-Base Theorem 71. (Modeling) Diversity of Species The number of species in a sample is given by S1n2 = a ln a1 + n ab , where n is the number of individuals in the sample, and a is a constant that indicates the diversity of species in the community. If a = 0.36, find S1n2 for each value of n. (Hint: S1n2 must be a whole number.) (a) 100 (b) 200 (c) 150 (d) 10 72. (Modeling) Diversity of Species (Refer to Exercise 71.) Find S1n2 if a changes to 0.88. Use the following values of n. (a) 50 (b) 100 (c) 250 73. (Modeling) Diversity of Species Suppose a sample of a community shows two species with 50 individuals each. Use the formula to find the measure of diversity H, where P1, P2, c Pn are the proportions of a sample belonging to each of n species. H= -3P1 log2 P1 + P2 log2 P2 + g+ Pn log2 Pn4 74. (Modeling) Diversity of Species (Refer to Exercise 73.) A virgin forest in northwestern Pennsylvania has 4 species of large trees with the following proportions of each: hemlock, 0.521; beech, 0.324; birch, 0.081; maple, 0.074. Use the formula to find the measure of diversity H to the nearest thousandth. 75. (Modeling) Global Temperature Increase In Example 7, we expressed the average global temperature increase T (in °F) as T1k2 = 1.03k ln C C0 , where C0 is the preindustrial amount of carbon dioxide, C is the current carbon dioxide level, and k is a constant. Arrhenius determined that 10 … k … 16 when C was double the value C0 . Use T1k2 to find the range of the rise in global temperature T (rounded to the nearest degree) that Arrhenius predicted. (Data from Clime, W., The Economics of Global Warming, Institute for International Economics, Washington, D.C.) 76. (Modeling) Global Temperature Increase (Refer to Exercise 75.) According to the IPCC, future increases in average global temperatures (in °F) can be modeled by T1C2 = 6.489 ln C 280 , where C is the concentration of atmospheric carbon dioxide (in ppm). C can be modeled by the function C1x2 = 35311.0062x - 1990, where x is the year. (Data from International Panel on Climate Change (IPCC).) (a) Write T as a function of x. (b) Using a graphing calculator, graph C1x2 and T1x2 on the interval [1990, 2275] using different coordinate axes. Describe the graphs. How are C and T related? (c) Approximate the slope of the graph of T. What does this slope represent? (d) Use graphing to estimate x and C1x2 when T1x2 = 10°F. 77. Age of Rocks Use the formula to estimate the age, in billions of years, of a rock sample having A K = 0.103. Round to the nearest hundredth. t = 11.26 * 1092 lnA1 + 8.33 AA KB B ln 2
RkJQdWJsaXNoZXIy NjM5ODQ=