# The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. begin{array}{|c|c|} hline text{Year} & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 hline text{Sales, S} & 898.2 & 1024.1 & 1366.8 & 1447.4 & 1822.1 & 2006.9 & 2268.9 hline end{array} a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t=4 corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property b=e^{ln b} to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011.

Question
Exponential models
The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&{2004}&{2005}&{2006}&{2007}&{2008}&{2009}&{2010}\backslash{h}{l}\in{e}\text{Sales, S}&{898.2}&{1024.1}&{1366.8}&{1447.4}&{1822.1}&{2006.9}&{2268.9}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with $$\displaystyle{t}={4}$$ corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property $$\displaystyle{b}={e}^{{{\ln{\ }}{b}}}$$ to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011. It is projected that sales in 2011 will be $2740 million. Do the predictions from the two models agree with this projection? Explain. ## Answers (1) 2021-01-20 Step 1 Exponential Regression Model Step 2 a) A scatter plot of the given data is displayed below: Step 3 b) Using the regression feature, we find that an exponential model for the data in base e is $$\displaystyle{f{{\left({x}\right)}}}={488.964}{e}^{{{0.158}{x}}}$$ Step 4 c) Using the regression feature, A logarithmic model of given data is given by $$\displaystyle{g{{\left({x}\right)}}}=\ -{1333.635}\ +\ {1515.088}\ {\ln{\ }}{\left({x}\right)}$$ Step 5 d) To predict sales in 2011 using the exponential model, find f(11) as follows: $$\displaystyle{f{{\left({11}\right)}}}={488.964}{e}^{{{0.158}{\left({11}\right)}}}\ \approx\ {2768.29}$$ To predict sales in 2011 using the logarithmic model, find g(11) as follows: $$\displaystyle{g{{\left({11}\right)}}}=\ -{1333.635}\ +\ {1515.088}\ {\ln{\ }}{\left({11}\right)}\ \approx\ {2299.39}$$ The exponential model closely agrees with the projected amount. The exponential model is only off by about 28.29 million while the logarithmic model is off by a significant 440 million. ### Relevant Questions asked 2020-11-08 The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected. Source: Centers for Disease Control and Prevention. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{amp, Cases}\backslash{h}{l}\in{e}{1995}&{a}\mp,\ {295}\backslash{h}{l}\in{e}{1997}&{a}\mp,\ {166}\backslash{h}{l}\in{e}{1999}&{a}\mp,\ {109}\backslash{h}{l}\in{e}{2001}&{a}\mp,\ {115}\backslash{h}{l}\in{e}{2003}&{a}\mp,\ {94}\backslash{h}{l}\in{e}{2005}&{a}\mp,\ {107}\backslash{h}{l}\in{e}{2007}&{a}\mp,\ {79}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ a) Plot the data on a graphing calculator, letting $$\displaystyle{t}={0}$$ correspond to the year 1995. b) Using the regression feature on your calculator, find a quadratic, a cubic, and an exponential function that models this data. c) Plot the three functions with the data on the same coordinate axes. Which function or functions best capture the behavior of the data over the years plotted? d) Find the number of cases predicted by all three functions for 20152015. Which of these are realistic? Explain. asked 2021-02-18 For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&{1}&{2}&{3}&{4}&{5}&{6}&{7}&{8}&{9}&{10}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{409.4}&{260.7}&{170.4}&{110.6}&{74}&{44.7}&{32.4}&{19.5}&{12.7}&{8.1}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ asked 2021-01-07 The U.S. Census Bureau publishes information on the population of the United States in Current Population Reports. The following table gives the resident U.S. population, in millions of persons, for the years 1990-2009. Forecast the U.S. population in the years 2010 and 2011 PSK\begin{array}{|c|c|} \hline \text{Year} & \text{Population (millions)} \\ \hline 1990 & 250 \\ \hline 1991 & 253\\ \hline 1992 & 257\\ \hline 1993 & 260\\ \hline 1994 & 263\\ \hline 1995 & 266\\ \hline 1996 & 269\\ \hline 1997 & 273\\ \hline 1998 & 276\\ \hline 1999 & 279\\ \hline 2000 & 282\\ \hline 2001 & 285\\ \hline 2002 & 288\\ \hline 2003 & 290\\ \hline 2004 & 293\\ \hline 2005 & 296\\ \hline 2006 & 299\\ \hline 2007 & 302\\ \hline 2008 & 304\\ \hline 2009 & 307\\ \hline \end{array}ZSK a) Obtain a scatterplot for the data. b) Find and interpret the regression equation. c) Mace the specified forecasts. asked 2021-01-31 The table gives the midyear population of Japan, in thousands, from 1960 to 2010. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{Population}\backslash{h}{l}\in{e}{1960}&{94.092}\backslash{h}{l}\in{e}{1965}&{98.883}\backslash{h}{l}\in{e}{1970}&{104.345}\backslash{h}{l}\in{e}{1975}&{111.573}\backslash{h}{l}\in{e}{1980}&{116.807}\backslash{h}{l}\in{e}{1985}&{120.754}\backslash{h}{l}\in{e}{1990}&{123.537}\backslash{h}{l}\in{e}{1995}&{125.327}\backslash{h}{l}\in{e}{2000}&{126.776}\backslash{h}{l}\in{e}{2005}&{127.715}\backslash{h}{l}\in{e}{2010}&{127.579}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose $$\displaystyle{t}={0}$$ to correspond to 1960 or 1980.] asked 2020-11-30 The table gives the midyear population of Norway, in thousands, from 1960 to 2010. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{Population}\backslash{h}{l}\in{e}{1960}&{3581}\backslash{h}{l}\in{e}{1965}&{3723}\backslash{h}{l}\in{e}{1970}&{3877}\backslash{h}{l}\in{e}{1975}&{4007}\backslash{h}{l}\in{e}{1980}&{4086}\backslash{h}{l}\in{e}{1985}&{4152}\backslash{h}{l}\in{e}{1990}&{4242}\backslash{h}{l}\in{e}{1995}&{4359}\backslash{h}{l}\in{e}{2000}&{4492}\backslash{h}{l}\in{e}{2005}&{4625}\backslash{h}{l}\in{e}{2010}&{4891}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 3500 from each of the population figures. Then, after obtaining a model from your calculator, add 3500 to get your final model. It might be helpful to choose $$\displaystyle{t}={0}$$ to correspond to 1960.] asked 2021-02-11 Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification. (a) (Exponential Model) This model is described by $$\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{e}}}{N}\ {\left({8.86}\right)}.$$ Solve (8.86) with the initial condition N(0) at time 0, and show that $$\displaystyle{r}_{{{e}}}$$ can be estimated from $$\displaystyle{r}_{{{e}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ {\left({8.87}\right)}$$ (b) (Logistic Growth) This model is described by $$\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{l}}}{N}\ {\left({1}\ -\ {\frac{{{N}}}{{{K}}}}\right)}\ {\left({8.88}\right)}$$ where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that $$\displaystyle{r}_{{{l}}}$$ can be estimated from $$\displaystyle{r}_{{{l}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{K}\ -\ {N}{\left({0}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ +\ {\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{K}\ -\ {N}{\left({t}\right)}}}}\right]}\ {\left({8.89}\right)}$$ for $$\displaystyle{N}{\left({t}\right)}\ {<}\ {K}.$$ (c) Assume that $$\displaystyle{N}{\left({0}\right)}={1}$$ and $$\displaystyle{N}{\left({10}\right)}={1000}.$$ Estimate $$\displaystyle{r}_{{{e}}}$$ and $$\displaystyle{r}_{{{l}}}$$ for both $$\displaystyle{K}={1001}$$ and $$\displaystyle{K}={10000}.$$ (d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of $$\displaystyle{\left[{r}\right]}$$ to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated. (e) Explain why the exponential model is a good approximation to the logistic model when $$\displaystyle\frac{{N}}{{K}}$$ is small compared with 1. asked 2020-12-25 Case: Dr. Jung’s Diamonds Selection With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring. After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung. 1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest? 2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why? 3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics? 4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase? 5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why? 1 6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics? 7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase? 8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why? 9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics? 10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase? asked 2020-11-06 A small grocer finds that the monthly sales y (in$) can be approximated as a function of the amount spent advertising on the radio $$x_1$$
(in $) and the amount spent advertising in the newspaper $$x_2$$ (in$) according to $$y=ax_1+bx_2+c$$
The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months.
$$\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline  2400 & { 800} & { 36,000} \\ \hline  2000 & { 500} & { 30,000} \\ \hline  3000 & { 1000} & { 44,000} \\ \hline\end{array}$$
a) Use the data to write a system of linear equations to solve for a, b, and c.
b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix.
c) Write the model $$y=ax_1+bx_2+c$$
d) Predict the monthly sales if the grocer spends $250 advertising on the radio and$500 advertising in the newspaper for a given month.
The table gives the number of active Twitter users worldwide, semiannually from 2010 to 2016. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Years since}&\text{January 1, 2010}&\text{Twitter user}&\text{(millions)}\backslash{h}{l}\in{e}{0}&{30}&{3.5}&{232}\backslash{h}{l}\in{e}{0.5}&{49}&{4.0}&{255}\backslash{h}{l}\in{e}{1.0}&{68}&{4.5}&{284}\backslash{h}{l}\in{e}{1.5}&{101}&{5.0}&{302}\backslash{h}{l}\in{e}{2.0}&{138}&{5.5}&{307}\backslash{h}{l}\in{e}{2.5}&{167}&{6.0}&{310}\backslash{h}{l}\in{e}{3.0}&{204}&{6.5}&{317}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Use a calculator or computer to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models.
Determine the algebraic modeling The personnel costs in the city of Greenberg were \$9,500,000 in 2009. In the recent past the personnel costs have increased at the rate of $$4.2\%$$ annually.
To model this, the city manager uses the function $$\displaystyle{C}{\left({t}\right)}={9.5}{\left({1}{042}\right)}^{t}\text{where}\ {C}{\left({t}\right)}$$ is the annual personnel costs,in millions of dollars, t years past 2009