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Chapter 3: Problem 79

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

Short Answer

Expert verified
The statement makes sense, since the negative growth rate \(k\) in an exponential function indicates a declining trend, which is consistent with the condition of population decline.

Step by step solution

01

Understand Population Growth and Decline

Population growth or decline is often modeled with exponential functions, given by \(P(t) = P_0 * e^{kt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(e\) is the base of the natural logarithm (approximately equal to 2.718), and \(k\) is the growth rate.
02

Understand Negative Growth Rate

Growth rate \(k\) is an important factor in the exponential function. When \(k > 0\), the population is increasing over time, hence, it's growth. When \(k < 0\), the population is decreasing over time, which can be described as negative growth or population decline.
03

Decision on Statement

Given that Russia's population is declining (as mentioned in the statement), if we are modeling this with an exponential function, it makes sense that the growth rate \(k\) is negative, since negative growth typically indicates a decline.

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Most popular questions from this chapter

What is the natural exponential function?

a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$ \log _{3} 81, \text { or } \log _{3} 9^{2} ? $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's linear regression option to obtain a model of the form \(y=a x+b\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

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