Chapter 5

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=x+2 y\) Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 4 \\ 2 x+y & \leq 4 \end{aligned} $$

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=125+0.0006 x\) Demand $$p=600-0.0002 x$$

Hair Products A hair product company sells three types of hair products for $$\$ 30$$, $$\$ 20$$, and $$\$ 10$$ per unit. In one year, the total revenue for the three products was $$\$ 800,000$$, which corresponded to the sale of 40,000 units. The company sold half as many units of the $$\$ 30$$ product as units of the $$\$ 20$$ product. How many units of each product were sold?

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ \left\\{\begin{array}{r} 5 b+10 a=11.7 \\ 10 b+30 a=25.6 \end{array}\right. $$

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{array}{l}-10 x+y=2 \\ -10 x+y=-3\end{array}\right.$$

Reasoning An objective function has a maximum value at the vertices \((0,14)\) and \((3,8)\). (a) Can you conclude that it also has a maximum value at the point \((1,12)\) ? Explain. (b) Can you conclude that it also has a maximum value at the point \((4,6)\) ? Explain. (c) Find another point that maximizes the objective function.

Think About It Under what circumstances are the consumer surplus and producer surplus equal for a pair of linear supply and demand equations? Explain.

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ \left\\{\begin{array}{r} 7 b+21 a=35.1 \\ 21 b+91 a=114.2 \end{array}\right. $$

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}y=x^{2}+3 x-1 \\ y=-x^{2}-2 x+2\end{array}\right.$$

Reasoning An objective function has a minimum value at the vertex \((20,0)\). Can you conclude that it also has a minimum value at the point \((0,0)\) ? Explain.

Think About It Under what circumstances is the consumer surplus greater than the producer surplus for a pair of linear supply and demand equations? Explain.

Acid Mixture A chemist needs 10 liters of a \(25 \%\) acid solution. The solution is to be mixed from three solutions whose acid concentrations are \(10 \%, 20 \%\), and \(50 \%\). How many liters of each solution should the chemist use to satisfy the following? (a) Use as little as possible of the \(50 \%\) solution. (b) Use as much as possible of the \(50 \%\) solution. (c) Use 2 liters of the \(50 \%\) solution.

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ \left\\{\begin{array}{r} 6 b+15 a=23.6 \\ 15 b+55 a=48.8 \end{array}\right. $$

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}y=-2 x^{2}+x-1 \\ y=x^{2}-2 x-1\end{array}\right.$$

Reasoning When solving a linear programming problem, you find that the objective function has a maximum value at more than one vertex. Can you assume that there are an infinite number of points that will produce the maximum value? Explain your reasoning.

Investment You plan to invest up to $$\$ 30,000$$ in two different interest- bearing accounts. Each account is to contain at least $$\$ 6000$$. Moreover, one account should have at least twice the amount that is in the other account. (a) Find a system of inequalities that describes the amounts that you can invest in each account, and (b) sketch the graph of the system.

You have a total of $$\$ 500,000$$ that is to be invested in (1) certificates of deposit, (2) municipal bonds, (3) blue-chip stocks, and (4) growth or speculative stocks. How much should be put in each type of investment? The certificates of deposit pay \(2.5 \%\) simple annual interest, and the municipal bonds pay \(10 \%\) simple annual interest. Over a five-year period, you expect the blue-chip stocks to return \(12 \%\) simple annual interest and the growth stocks to return \(18 \%\) simple annual interest. You want a combined annual return of \(10 \%\) and you also want to have only one-fourth of the portfolio invested in stocks.

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ (0,4),(1,3),(1,1),(2,0) $$

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}x-y+3=0 \\ x^{2}-4 x+7=y\end{array}\right.$$

Concert Ticket Sales Two types of tickets are to be sold for a concert. One type costs $$\$ 20$$ per ticket and the other type costs $$\$ 30$$ per ticket. The promoter of the concert must sell at least 20,000 tickets, including at least 8000 of the $$\$ 20$$ tickets and at least 5000 of the $$\$ 30$$ tickets. Moreover, the gross receipts must total at least $$\$ 480,000$$ in order for the concert to be held. (a) Find a system of inequalities describing the different numbers of tickets that must be sold, and (b) sketch the graph of the system.

You have a total of $$\$ 500,000$$ that is to be invested in (1) certificates of deposit, (2) municipal bonds, (3) blue-chip stocks, and (4) growth or speculative stocks. How much should be put in each type of investment? The certificates of deposit pay \(3 \%\) simple annual interest, and the municipal bonds pay \(10 \%\) simple annual interest. Over a five-year period, you expect the blue-chip stocks to return \(12 \%\) simple annual interest and the growth stocks to return \(15 \%\) simple annual interest. You want a combined annual return of \(10 \%\) and you also want to have only one-fourth of the portfolio invested in stocks.

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ (1,0),(2,0),(3,0),(3,1),(4,1),(4,2),(5,2),(6,2) $$

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}x-y=3 \\ x-y^{2}=1\end{array}\right.$$

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 20 units of calcium, 10 units of iron, and 15 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 20 units of vitamin \(\mathrm{B}\). The minimum daily requirements for the diet are 400 units of calcium, 250 units of iron, and 220 units of vitamin B. (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 18 ounces of food \(\mathrm{X}\) and \(3.5\) ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{X}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?

Classic Cars The numbers of cars \(y\) sold at BarrettJackson Collector Car Auction in Scottsdale in the years 2003 to 2007 are shown in the table. (Source: BarrettJackson Auction Company) $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline t & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Cars, } y & 655 & 727 & 877 & 1105 & 1271 \\ \hline \end{array} $$ (a) Solve the following system for \(a\) and \(b\) to find the least squares regression line \(y=a t+b\) for the data. Let \(t\) represent the year, with \(t=0\) corresponding to 2003 . \(\left\\{\begin{array}{c}5 b+10 a=4635 \\ 10 b+30 a=10,880\end{array}\right.\) (b) Use a graphing utility to graph the regression line and estimate the number of cars that will be sold in 2009 . (c) Use the regression feature of a graphing utility to find a linear model for the data. Compare this model with the one you found in part (a).

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}y=e^{x} \\ x-y+1=0\end{array}\right.$$

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 12 units of calcium, 10 units of iron, and 20 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B. The minimum daily requirements for the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin \(\mathrm{B}\). (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 10 ounces of food \(\mathrm{X}\) and 12 ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{Y}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?

Atmosphere The concentration \(y\) (in parts per million) of carbon dioxide in the atmosphere is measured at the Mauna Loa Observatory in Hawaii. The greatest monthly carbon dioxide concentrations for the years 2002 to 2006 are shown in the table. (Source: Scripps CO2 Program) $$ \begin{array}{|c|c|c|} \hline \text { Year } & t & \text { Concentration, } y \\ \hline 2002 & 0 & 375.55 \\ \hline 2003 & 1 & 378.35 \\ \hline 2004 & 2 & 380.63 \\ \hline 2005 & 3 & 382.26 \\ \hline 2006 & 4 & 384.92 \\ \hline \end{array} $$ (a) Solve the following system for \(a\) and \(b\) to find the least squares regression line \(y=a t+b\) for the data. Let \(t\) represent the year, with \(t=0\) corresponding to 2002 . \(\left\\{\begin{array}{r}5 b+10 a=1901.71 \\ 10 b+30 a=3826.07\end{array}\right.\) (b) Use a graphing utility to graph the regression line and predict the largest monthly carbon dioxide concentration in 2012 . (c) Use the regression feature of a graphing utility to find a linear model for the data. Compare this model with the one you found in part (a).

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}y=\sqrt{x} \\ y=x\end{array}\right.$$

Health A person's maximum heart rate is \(220-x\), where \(x\) is the person's age in years for \(20 \leq x \leq 70\). When a person exercises, it is recommended that the person strive for a heart rate that is at least \(50 \%\) of the maximum and at most \(75 \%\) of the maximum. (Source: American Heart Association) (a) Write a system of inequalities that describes the exercise target heart rate region. Let \(y\) represent a person's heart rate. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the system and interpret their meanings in the context of the problem.

Reasoning Design a system of two linear equations with infinitely many solutions. Solve the system algebraically and explain how the solution indicates that there are infinitely many solutions.

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}4 x^{2}-y^{2}-32 x-2 y=-59 \\ 2 x+y-7=0\end{array}\right.$$

Peregrine Falcons The numbers of nesting pairs \(y\) of peregrine falcons in Yellowstone National Park from 2001 to 2005 can be approximated by the linear model \(y=3.4 t+13, \quad 1 \leq t \leq 5\) where \(t\) represents the year, with \(t=1\) corresponding to 2001\. (Sounce: Yellowstone Bird Report 2005) (a) The total number of nesting pairs during this five-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 3.4 t+13 \\ y \geq 0 \\ t \geq 0.5 \\\ t \leq 5.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total number of nesting pairs.

Reasoning Design a system of two linear equations with no solution. Solve the system algebraically and explain how the solution indicates that there is no solution.

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}x^{2}+y^{2}=8 \\ y=x^{2}+4\end{array}\right.$$

Computers The sales \(y\) (in billions of dollars) for Dell Inc. from 1996 to 2005 can be approximated by the linear model \(y=5.07 t-22.4, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Dell Inc.) (a) The total sales during this ten-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 5.07 t-22.4 \\ y \geq 0 \\ t \geq 5.5 \\ t \leq 15.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total sales.

Sailboats The total numbers \(y\) (in thousands) of sailboats purchased in the United States in the years 2001 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to \(2003 .\) (Source: National Marine Manufacturers Association) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { Number, } y \\ \hline-2 & 18.6 \\ \hline-1 & 15.8 \\ \hline 0 & 15.0 \\ \hline 1 & 14.3 \\ \hline 2 & 14.4 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{aligned} 5 c &+10 a=78.1 \\\ 10 b &=-9.9 \\ 10 c &+34 a=162.1 \end{aligned}\right.\) (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).

Think About It For the system below, find the value(s) of \(k\) for which the system is (a) inconsistent and (b) consistent (dependent). Explain how you found your answers. \(\left\\{\begin{array}{r}3 x-12 y=9 \\ x-4 y=k\end{array}\right.\)

Break-Even Analysis, find the sales necessary to break even \((R=C)\) for the cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round your answer to the nearest whole unit.) $$C=8650 x+250,000 ; R=9950 x$$

Write a system of inequalities whose graphed solution set is a right triangle.

Genetically Modified Soybeans The global areas \(y\) (in millions of hectares) of genetically modified soybean crops planted in the years 2002 to 2006 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to 2004. (Source: ISAAA, Clive James, 2006) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { Area, } y \\ \hline-2 & 36.5 \\ \hline-1 & 41.4 \\ \hline 0 & 48.4 \\ \hline 1 & 54.4 \\ \hline 2 & 58.6 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{l}5 c+10 a=239.3 \\\ 10 b \quad=57.2 \\ 10 c+34 a=476.2\end{array}\right.\) (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).

Break-Even Analysis, find the sales necessary to break even \((R=C)\) for the cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round your answer to the nearest whole unit.) $$C=5.5 \sqrt{x}+10,000 ; R=3.29 x$$

Write a system of inequalities whose graphed solution set is an isosceles triangle.

Federal Debt The values of the federal debt of the United States as percents of the Gross Domestic Product (GDP) for the years 2001 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to 2002. (Source: U.S. Office of Management and Budget) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { \% of GDP } \\ \hline-1 & 57.4 \\ \hline 0 & 59.7 \\ \hline 1 & 62.6 \\ \hline 2 & 63.7 \\ \hline 3 & 64.3 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}5 c+5 b+15 a=307.7 \\\ 5 c+15 b+35 a=325.5 \\ 15 c+35 b+99 a=953.5\end{array}\right.\) (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 federal debt as a percent of the GDP in 2007 .

Break-Even Analysis, find the sales necessary to break even \((R=C)\) for the cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round your answer to the nearest whole unit.) $$C=2.65 x+350,000 ; R=4.15 x$$

Writing Explain the difference between the graphs of the inequality \(x \leq 4\) on the real number line and on the rectangular coordinate system.

Revenues Per Share The revenues per share (in dollars) for Panera Bread Company for the years 2002 to 2006 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to \(2003 .\) (Source: Panera Bread Company) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { Revenues per share } \\ \hline-1 & 9.47 \\ \hline 0 & 11.85 \\ \hline 1 & 15.72 \\ \hline 2 & 20.49 \\ \hline 3 & 26.11 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}5 c+5 b+15 a=83.64 \\\ 5 c+15 b+35 a=125.56 \\ 15 c+35 b+99 a=342.14\end{array}\right.\) (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 revenues per share in 2008 and \(2009 .\)

Break-Even Analysis, find the sales necessary to break even \((R=C)\) for the cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round your answer to the nearest whole unit.) $$C=0.08 x+50,000 ; R=0.25 x$$

Graphical Reasoning Two concentric circles have radii \(x\) and \(y\), where \(y>x .\) The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

MAKE A DECISION: STOPPING DISTANCE In testing of the new braking system of an automobile, the speed (in miles per hour) and the stopping distance (in feet) were recorded in the table below. $$ \begin{array}{|c|c|} \hline \text { Speed, } x & \text { Stopping distance, } y \\ \hline 30 & 54 \\ \hline 40 & 116 \\ \hline 50 & 203 \\ \hline 60 & 315 \\ \hline 70 & 452 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}5 c+250 b+13,500 a=1140 \\ 250 c+13,500 b+775,000 a=66,950 \\ 13,500 c+775,000 b+46,590,000 a=4,090,500\end{array}\right.\) (b) Use the regression feature of a graphing utility to check your answer to part (a). (c) A car design specification requires the car to stop within 520 feet when traveling 75 miles per hour. Does the new braking system meet this specification?