Chapter 5

Break-Even Analysis You invest \(\$ 18,000\) in equipment to make CDs. The CDs can be produced for \(\$ 1.95\) each and will be sold for \(\$ 13.95\) each. How many CDs must you sell to break even?

Sound Recordings The percents of sound recordings purchased over the Internet (not including digital downloads) in the years 1999 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to 2000. (Source: The Recording Industry Association of America) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \begin{array}{c} \text { Percent of sound } \\ \text { recordings, } y \end{array} \\ \hline-1 & 2.4 \\ \hline 0 & 3.2 \\ \hline 1 & 2.9 \\ \hline 2 & 3.4 \\ \hline 3 & 5.0 \\ \hline 4 & 5.9 \\ \hline 5 & 8.2 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}7 c+14 b+56 a=31.0 \\\ 14 c+56 b+224 a=86.9 \\ 56 c+224 b+980 a=363.3\end{array}\right.\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. Compare the quadratic model with the model found in part (a). (c) Use either model to predict the percent of Internet sales in 2008 . Does your result seem reasonable? Explain.

Break-Even Analysis You invest \(\$ 3000\) in a fishing lure business. A lure costs \(\$ 1.06\) to produce and will be sold for \(\$ 5.86 .\) How many lures must you sell to break even?

Reasoning Is it possible for a square linear system to have no solution? Explain.

Comparing Populations From 1995 to 2005, the population of Kentucky grew more slowly than that of Colorado. Models that represent the populations of the two states are given by \(\left\\{\begin{array}{ll}P=27.9 t+3757 & \text { Kentucky } \\ P=86.1 t+3425 & \text { Colorado }\end{array}\right.\) where \(P\) is the population (in thousands) and \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use the models to estimate when the population of Colorado first exceeded the population of Kentucky. (Source: U.S. Census Bureau)

Reasoning Is it possible for a square linear system to have infinitely many solutions? Explain.

Comparing Populations From 1995 to 2005, the population of Maryland grew more slowly than that of Arizona. Models that represent the populations of the two states are given by \(\left\\{\begin{array}{ll}P=55.6 t+4771 & \text { Maryland } \\ P=145.9 t+3703 & \text { Arizona }\end{array}\right.\) where \(P\) is the population (in thousands) and \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use the models to estimate when the population of Arizona first exceeded the population of Maryland. (Source: U.S. Census Bureau)

Body Mass Index Body mass index (BMI) is a measure of body fat based on height and weight. The 75 th percentile BMI for females, ages 9 to 20 , grew more slowly than that of males of the same age range. Models that represent the 75 th percentile BMI for males and females, ages 9 to 20 , are given by \(\left\\{\begin{array}{ll}B=0.73 a+11 & \text { Males } \\ B=0.61 a+12.8 & \text { Females }\end{array}\right.\) where \(B\) is the \(\mathrm{BMI}\left(\mathrm{kg} / \mathrm{m}^{2}\right)\) and \(a\) represents the age, with \(a=9\) corresponding to 9 years old. Use a graphing utility to determine whether the BMI for males will exceed the BMI for females. (Source: National Center for Health Statistics)

Clothing Sales From 1996 to 2005, the sales of Abercrombie \& Fitch Company grew faster than those of Timberland Company. Models that represent the sales of the two companies are given by \(\left\\{\begin{array}{ll}S=235.1 t-1126 & \text { Abercrombie } \& \text { Fitch Company } \\ S=97.7 t+88 & \text { Timberland Company }\end{array}\right.\) where \(S\) is the sales (in millions) and \(t\) represents the year, with \(t=6\) corresponding to 1996 . Use a graphing utility to determine whether the sales of Abercrombie \& Fitch Company will exceed the sales of Timberland Company.

A total of $$\$ 35,000$$ is invested in two funds paying \(8.5 \%\) and \(12 \%\) simple interest. The total annual interest is $$\$ 3675$$. How much is invested at each rate?

A total of $$\$ 35,000$$ is invested in two funds paying \(8 \%\) and \(10.5 \%\) simple interest. The total annual interest is $$\$ 3275 .$$ How much is invested at each rate?

Job Choices You are offered two different jobs. Company A offers an annual salary of \(\$ 30,000\) plus a year-end bonus of \(2.5 \%\) of your total sales. Company \(\mathrm{B}\) offers a salary of \(\$ 24,000\) plus a year-end bonus of \(6.5 \%\) of your total sales. What is the amount you must sell in one year to eam the same salary working for either company?

Camping You are choosing between camping outfitters. Outfitter A charges a reservation fee of $$\$ 150$$ plus a daily guide fee of $$\$ 70 .$$Outfitter \(\mathrm{B}\) charges a reservation fee of $$\$ 75$$ plus a daily guide fee of $$\$ 90 .$$ Estimate when the cost of Outfitter A equals the cost of Outfitter \(\mathrm{B}\).

Financial Aid The average award for Federal Pell Grants and Federal Perkins Loans from 1995 to 2005 can be approximated by \(=-2.051 t^{3}+56.87 t^{2}-376.7 t+2238\) Federal Pell Grant \(=-1.810 t^{3}+56.64 t^{2}-476.4 t+2711\) Federal Perkins Loan where \(A\) is the award (in dollars) and \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use a graphing utility to determine whether Federal Perkins Loan awards will exceed Federal Pell Grant awards. Do you think these models will continue to be accurate? Explain your reasoning. (Source: U.S. Department of Education)

SAT or ACT? The number of participants in SAT and ACT testing from 1995 to 2005 can be approximated by \(\left\\{\begin{array}{ll}y=0.68 t^{2}+28.1 t+903 & \text { SAT } \\ y=-0.485 t^{3}+14.88 t^{2}-115.1 t+1201 & \text { ACT }\end{array}\right.\) where \(y\) is the number of participants (in thousands) and \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use a graphing utility to determine whether the number of participants in ACT testing will exceed the number of participants in SAT testing. Do you think these models will continue to be accurate? Explain your reasoning. (Sounce: College Board; ACT, Inc.)