Chapter 5

Optimal Profit A company makes two models of doghouses. The times (in hours) required for assembling, painting, and packaging are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2.5 & 3 \\ \hline \text { Painting } & 2 & 1 \\ \hline \text { Packaging } & 0.75 & 1.25 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 2500 hours, and 1500 hours, respectively. The profits per unit are $$\$ 60$$ for model \(\mathrm{A}\) and $$\$ 75$$ for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?

Graph the solution set of the system of inequalities. $$\left\\{\begin{aligned} x^{2}+y & \leq 4 \\ y & \geq 2 x \\ x & \geq-1 \end{aligned}\right.$$

Write three ordered triples of the given form. $$(3 a, 5-a, a)$$

Fuel Mixture Five hundred gallons of 89 -octane gasoline is obtained by mixing 87 -octane gasoline with 92 -octane gasoline. How much of each type of gasoline is required to obtain the specified mixture? (Octane ratings can be interpreted as percents.)

Solve the system graphically. $$\left\\{\begin{aligned}-x+2 y &=1 \\ x-y &=2 \end{aligned}\right.$$

Optimal Revenue An accounting firm charges $$\$ 2500$$ for an audit and $$\$ 350$$ for a tax return. Research and available resources have indicated the following constraints. \- The firm has 900 hours of staff time available each week. \- The firm has 155 hours of review time available each week. \- Each audit requires 75 hours of staff time and 10 hours of review time. \- Each tax return requires \(12.5\) hours of staff time and \(2.5\) hours of review time. What numbers of audits and tax returns will bring in an optimal revenue?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \leq e^{x} \\ y \geq \ln x \\ x \geq \frac{1}{2} \\\ x \leq 2\end{array}\right.$$

Investment Portfolio A total of \(\$ 25,000\) is invested in two corporate bonds that pay \(9.5 \%\) and \(14 \%\) simple interest. The total annual interest is \(\$ 3050 .\) How much is invested in each bond?

Solve the system graphically. $$\left\\{\begin{array}{r}x+y=4 \\ x^{2}+y^{2}-4 x=0\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \leq e^{-x^{2} / 2} \\ y \geq 0 \\ x \geq-1 \\\ x \leq 0\end{array}\right.$$

Investment Portfolio A total of $$\$ 50,000$$ is invested in two municipal bonds that pay \(6.75 \%\) and \(8.25 \%\) simple interest. The total annual interest is $$\$ 3900$$. How much is invested in each bond?

Solve the system graphically. $$\left\\{\begin{array}{r}-x+y=3 \\ x^{2}-6 x-27+y^{2}=0\end{array}\right.$$

Ticket Sales You are the manager of a theater. On Saturday morning you are going over the ticket sales for Friday evening. A total of 740 tickets were sold. The tickets for adults and children sold for $$\$ 8.50$$ and $$\$ 4.00$$, respectively, and the total receipts for the performance were $$\$ 4688$$. However, your assistant manager did not record how many of each type of ticket were sold. From the information you have, can you determine how many of each type were sold? Explain your reasoning.

Solve the system graphically. $$\left\\{\begin{aligned} x-y+3 &=0 \\ x^{2}-4 x+7 &=y \end{aligned}\right.$$

Shoe Sales You are the manager of a shoe store. On Sunday morning you are going over the receipts for the previous week's sales. A total of 320 pairs of cross-training shoes were sold. One style sold for $$\$ 56.95$$ and the other sold for $$\$ 72.95 .$$ The total receipts were $$\$ 21,024$$. The cash register that was supposed to keep track of the number of each type of shoe sold malfunctioned. Can you recover the information? If so, how many of each type were sold?

Investments An investor has up to $$\$ 250,000$$ to invest in two types of investments. Type A investments pay \(7 \%\) annually and type \(\mathrm{B}\) pay \(12 \%\) annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-fourth of the total portfolio is to be allocated to type A investments and at least one-fourth is to be allocated to type \(\mathrm{B}\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

Supply and Demand, find the point of equilibrium for the pair of demand and supply equations. Supply \(p=22+0.00001 x\) Demand $$p=56-0.0001 x$$

Solve the system graphically. $$\left\\{\begin{aligned} 7 x+8 y &=24 \\ x-8 y &=8 \end{aligned}\right.$$

Investments An investor has up to $$\$ 450,000$$ to invest in two types of investments. Type A investments pay \(8 \%\) annually and type \(B\) pay \(14 \%\) annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth is to be allocated to type \(\mathrm{B}\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

Supply and Demand, find the point of equilibrium for the pair of demand and supply equations. Supply \(p=15+0.00004 x\) Demand $$p=60-0.00001 x$$

Solve the system graphically. $$\left\\{\begin{array}{r}x-y=0 \\ 5 x-2 y=6\end{array}\right.$$

Optimal Profit A company makes two models of a patio furniture set. The times for assembling, finishing, and packaging model \(\mathrm{A}\) are 3 hours, \(2.5\) hours, and \(0.6\) hour, respectively. The times for model \(\mathrm{B}\) are \(2.75\) hours, 1 hour, and \(1.25\) hours. The total times available for assembling, finishing, and packaging are 3000 hours, 2400 hours, and 1200 hours, respectively. The profit per unit for model \(\mathrm{A}\) is $$\$ 100$$ and the profit per unit for model \(\mathrm{B}\) is $$\$ 85 .$$ What is the optimal production level for each model? What is the optimal profit?

Supply and Demand, find the point of equilibrium for the pair of demand and supply equations. Supply \(p=80+0.00001 x\) Demand $$p=140-0.00002 x$$

Solve the system graphically. $$\left\\{\begin{array}{l}3 x-2 y=0 \\ x^{2}-y^{2}=4\end{array}\right.$$

Optimal Profit A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model \(\mathrm{A}\) are 3 hours, 3 hours, and \(0.8\) hour, respectively. The times for model B are 4 hours, \(2.5\) hours, and \(0.4\) hour. The total times available for assembling, finishing, and packaging are 6000 hours, 4200 hours, and 950 hours, respectively. The profits per unit are \(\$ 300\) for model \(A\) and $$\$ 375$$ for model \(B\). What is the optimal production level for each model? What is the optimal profit?

Supply and Demand, find the point of equilibrium for the pair of demand and supply equations. Supply \(p=225+0.0005 x\) Demand $$P=400-0.0002 x$$

Solve the system graphically. $$\left\\{\begin{array}{r}2 x-y+3=0 \\ x^{2}+y^{2}-4 x=0\end{array}\right.$$

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: \(z=2.5 x+y\) Constraints: \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \\ 5 x+2 y & \leq 10 \end{aligned}\)

Furniture Production A furniture company produces tables and chairs. Each table requires 2 hours in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. Each chair requires \(1 \frac{1}{2}\) hours in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. The company's assembly center is available 18 hours per day, and its finishing center is available 15 hours per day. Let \(x\) and \(y\) be the numbers of tables and chairs produced per day, respectively. (a) Find a system of inequalities describing all possible production levels, and (b) sketch the graph of the system.

Restaurants The total sales \(y\) (in billions of dollars) for fast-food and full-service restaurants for the years 1999 to 2005 are shown in the table. (Source: National Restaurant Association $$ \begin{array}{|c|c|c|} \hline \text { Year } & \text { Fast-food } & \text { Full-service } \\ \hline 1999 & 103.0 & 125.4 \\ \hline 2000 & 107.1 & 133.8 \\ \hline 2001 & 111.6 & 139.9 \\ \hline 2002 & 115.1 & 141.9 \\ \hline 2003 & 120.5 & 148.3 \\ \hline 2004 & 129.4 & 157.0 \\ \hline 2005 & 135.6 & 164.9 \\ \hline \end{array} $$ (a) Use a spreadsheet software program to create a scatter plot of the data for fast-food sales and use the regression feature to find a linear model. Let \(x\) represent the year, with \(x=9\) corresponding to 1999. Repeat the procedure for the data for full-service sales. (b) Assuming that the amounts for the given 7 years are representative of future years, will fast-food sales ever equal full-service sales?

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{array}{l}y=-5 x+1 \\ y=x+3\end{array}\right.$$

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: \(z=x+y\) Constraints: \(x \geq 0\) \(y \geq 0\) \(-x+y \leq 1\) \(-x+2 y \leq 4\)

Kayak Inventory A store sells two models of kayaks. Because of the demand, it is necessary to stock at least twice as many units of model \(\mathrm{A}\) as units of model \(\mathrm{B}\). The costs to the store for the two models are \(\$ 500\) and \(\$ 700\), respectively. The management does not want more than \(\$ 30,000\) in kayak inventory at any one time, and it wants at least six model A kayaks and three model B kayaks in inventory at all times. (a) Find a system of inequalities describing all possible inventory levels, and (b) sketch the graph of the system.

Prescriptions The numbers of prescriptions \(y\) (in thousands) filled at two pharmacies in the years 2002 to 2008 are shown in the table. $$ \begin{array}{|c|c|c|} \hline \text { Year } & \text { Pharmacy A } & \text { Pharmacy B } \\ \hline 2002 & 18.1 & 19.5 \\ \hline 2003 & 18.6 & 19.9 \\ \hline 2004 & 19.2 & 20.4 \\ \hline 2005 & 19.6 & 20.8 \\ \hline 2006 & 20.0 & 21.1 \\ \hline 2007 & 20.4 & 21.4 \\ \hline 2008 & 21.3 & 22.0 \\ \hline \end{array} $$ (a) Use a spreadsheet software program to create a scatter plot of the data for pharmacy A and use the regression feature to find a linear model. Let \(x\) represent the year, with \(x=2\) corresponding to 2002 . Repeat the procedure for the data for pharmacy \(\mathrm{B}\). (b) Assuming the amounts for the given 7 years are representative of future years, will the number of prescriptions filled at pharmacy A ever exceed the number of prescriptions filled at pharmacy \(\mathrm{B}\) ?

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{aligned}-\frac{1}{2} x+y &=-1 \\ 7 x+y &=2 \end{aligned}\right.$$

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: \(\begin{array}{rr}x & \geq 0 \\ y & \geq 0 \\ x & \leq 10 \\ x+y & \leq 7\end{array}\)

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=22+0.00001 x\) Demand $$p=56-0.0001 x$$

Investment A real estate company borrows $$\$ 1,500,000 .$$ Some of the money is borrowed at \(7 \%\), some at \(8 \%\), and some at \(10 \%\) simple annual interest. How much is borrowed at each rate when the total annual interest is $$\$ 117,000$$ and the amount borrowed at \(8 \%\) is the same as the amount borrowed at \(10 \%\) ?

Supply and Demand The supply and demand equations for a small LCD television are given by \(\left\\{\begin{array}{ll}p+0.53 x=1542 & \text { Demand } \\\ p-0.37 x=300 & \text { Supply }\end{array}\right.\) where \(p\) is the price (in dollars) and \(x\) represents the number of televisions. For how many units will the quantity demanded equal the quantity supplied? What price corresponds to this value?

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{array}{l}y=x^{2}+2 x-1 \\ y=2 x+5\end{array}\right.$$

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: \(z=x+y\) Constraints: \(x \geq 0\) \(y \geq 0\) \(-x+y \leq 1\) \(-3 x+y \geq 3\)

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=15+0.00004 x\) Demand $$p=60-0.00001 x$$

Investment A clothing company borrows $$\$ 700,000$$. Some of the money is borrowed at \(8 \%\), some at \(9 \%\), and some at \(10 \%\) simple annual interest. How much is borrowed at each rate when the total annual interest is $$\$ 60,500$$ and the amount borrowed at \(8 \%\) is three times the amount borrowed at \(10 \%\) ?

Supply and Demand The supply and demand equations for a microscope are given by \(\left\\{\begin{array}{ll}p+0.85 x=650 & \text { Demand } \\ p-0.4 x=75 & \text { Supply }\end{array}\right.\) where \(p\) is the price (in dollars) and \(x\) represents the number of microscopes. For how many units will the quantity demanded equal the quantity supplied? What price corresponds to this value?

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{aligned} x^{2}+3 x+y &=4 \\ 3 x+y &=-5 \end{aligned}\right.$$

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. \(z=3 x+4 y\) Constraints. \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \leq 1 \\ 2 x+y & \leq 4 \end{aligned}\)

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=80+0.00001 x\) Demand $$p=140-0.00002 x$$

Candles A candle company sells three types of candles for $$\$ 15$$, $\$ 10$$, and $$\$ 5$$ per unit. In one year, the total revenue for the three products was $$\$ 550,000$$, which corresponded to the sale of 50,000 units. The company sold half as many units of the $$\$ 15$$ candles as units of the $$\$ 10$$ candles. How many units of each type of candle 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.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ \left\\{\begin{array}{r} 5 b+10 a=20.2 \\ 10 b+30 a=50.1 \end{array}\right. $$

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{array}{l}y=x^{2}+3 x+7 \\ y=-x^{2}-3 x+1\end{array}\right.$$