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Chapter 4: Problem 74

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)

Short Answer

Expert verified
The statement does not make sense. The comparison of population sizes given the compounded growth over time does not result in a simple linear relationship (e.g., 3% growth resulting in three times the population of 1% growth). Exponential growth results in a much more significant population size as the growth rate increases.

Step by step solution

01

Set-up the Population Growth Formula

Use the exponential growth formula to calculate the final population after 100 years for both growth rates. Set \(P_0\) (initial population) as \(x\) for simplicity.
02

Calculate the Population for a 3% Growth Rate

Substitute \(r = 0.03\) and \(t = 100\) into the exponential growth formula. Simplify to get \(P = x e^{0.03*100}\).
03

Calculate the Population for a 1% Growth Rate

Substitute \(r = 0.01\) and \(t = 100\) into the exponential growth formula. Simplify to get \(P = x e^{0.01*100}\).
04

Compare the Two Calculated Populations

At this point, compare the populations obtained in steps 2 and 3. Is the population for a 3% growth rate three times greater than that for a 1% growth rate?

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Most popular questions from this chapter

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By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)f(x)=1.2 \ln x+15.7\( where \)f(x)\( is the percentage of the U.S. gross domestic product going toward health care \)x\( years after \)2006 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)2008 .\( Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \)18.6 \%$ of the U.S. gross domestic product go toward health care? Round to the nearest year.

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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } x=\frac{1}{k} \ln y, \text { then } y=e^{k x} $$

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use the values of \(r\) in Exercises \(66-69\) to select the two model= of best fit. Use each of these models to predict by which yeathe U.S. population will reach 335 million. How do these answers compare to the year we found in Example \(1,\) namel \(=\) \(2020 ?\) If you obtained different years, how do you account fo this difference?
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