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Chapter 4: Problem 22
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$e^{x+4}=\frac{1}{e^{2 x}}$$
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We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years affer 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.1 billion for \(2000 ?\)
Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\log x, g(x)=-\log x$$
Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect? $$ \text { Solve: } 5(2 x-3)-4 x=9 $$
You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?
Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The logistic growth function $$ f(t)=\frac{100,000}{1+5000 e^{-t}} $$ describes the number of people, \(f(t),\) who have become ill with influenza \(t\) weeks after its initial outbreak in a particular community. a. How many people became ill with the flu when the epidemic began? b. How many people were ill by the end of the fourth week? c. What is the limiting size of the population that becomes ill?
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