Appendix F: Systems of Inequalities

Transcription

1 A0 Appendi F Sstems of Inequalities Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit The statements < and are inequalities in two variables. An ordered pair a, b is a solution of an inequalit in and if the inequalit is true when a and b are substituted for and, respectivel. The graph of an inequalit is the collection of all solutions of the inequalit. To sketch the graph of an inequalit, begin b sketching the graph of the corresponding equation. The graph of the equation will normall separate the plane into two or more regions. In each such region, one of the following must be true.. All points in the region are solutions of the inequalit.. No point in the region is a solution of the inequalit. So, ou can determine whether the points in an entire region satisf the inequalit b simpl testing one point in the region. What ou should learn Sketch graphs of inequalities in two variables. Solve sstems of inequalities. Use sstems of inequalities in two variables to model and solve real-life problems. Wh ou should learn it Sstems of inequalities in two variables can be used to model and solve real-life problems. For instance, Eercise 8 on page A5 shows how to use a sstem of inequalities to analze the compositions of dietar supplements. Sketching the Graph of an Inequalit in Two Variables. Replace the inequalit sign with an equal sign and sketch the graph of the corresponding equation. Use a dashed line for < or > and a solid line for or. (A dashed line means that all points on the line or curve are not solutions of the inequalit. A solid line means that all points on the line or curve are solutions of the inequalit.). Test one point in each of the regions formed b the graph in Step. If the point satisfies the inequalit, shade the entire region to denote that ever point in the region satisfies the inequalit. Eample Sketching the Graph of an Inequalit Sketch the graph of b hand. Begin b graphing the corresponding equation, which is a parabola, as shown in Figure F.. B testing a point above the parabola 0, 0 and a point below the parabola 0,, ou can see that 0, 0 satisfies the inequalit because 0 0 and that 0, does not satisf the inequalit because > 0. So, the points that satisf the inequalit are those ling above and those ling on the parabola. Now tr Eercise 9. Figure F. The inequalit in Eample is a nonlinear inequalit in two variables. Most of the following eamples involve linear inequalities such as a b < c ( a and b are not both zero). The graph of a linear inequalit is a half-plane ling on one side of the line a b c.

2 Appendi F. Solving Sstems of Inequalities A07 Eample Sketching the Graphs of Linear Inequalities Sketch the graph of each linear inequalit. a. > b. a. The graph of the corresponding equation is a vertical line. The points that satisf the inequalit > are those ling to the right of (but not on) this line, as shown in Figure F.. b. The graph of the corresponding equation is a horizontal line. The points that satisf the inequalit are those ling below (or on) this line, as shown in Figure F.. = > = TECHNOLOGY TIP A graphing utilit can be used to graph an inequalit. For instance, to graph, enter and use the shade feature of the graphing utilit to shade the correct part of the graph. You should obtain the graph shown below. 9 9 For instructions on how to use the shade feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Figure F. Figure F. Eample Now tr Eercise 5. Sketching the Graph of a Linear Inequalit Sketch the graph of <. The graph of the corresponding equation is a line, as shown in Figure F.. Because the origin 0, 0 satisfies the inequalit, the graph consists of the half-plane ling above the line. (Tr checking a point below the line. Regardless of which point below the line ou choose, ou will see that it does not satisf the inequalit.) Now tr Eercise 7. < (0, 0) = Figure F. To graph a linear inequalit, it can help to write the inequalit in slopeintercept form. For instance, b writing < in Eample in the form > ou can see that the solution points lie above the line or, as shown in Figure F..

3 A08 Appendi F Sstems of Inequalities Sstems of Inequalities Man practical problems in business, science, and engineering involve sstems of linear inequalities. A solution of a sstem of inequalities in and is a point, that satisfies each inequalit in the sstem. To sketch the graph of a sstem of inequalities in two variables, first sketch the graph of each individual inequalit (on the same coordinate sstem) and then find the region that is common to ever graph in the sstem. For sstems of linear inequalities, it is helpful to find the vertices of the solution region. Eample Solving a Sstem of Inequalities Sketch the graph (and label the vertices) of the solution set of the sstem. < Inequalit > Inequalit Inequalit The graphs of these inequalities are shown in Figures F., F., and F., respectivel. The triangular region common to all three graphs can be found b superimposing the graphs on the same coordinate sstem, as shown in Figure F.5. To find the vertices of the region, solve the three sstems of corresponding equations obtained b taking pairs of equations representing the boundaries of the individual regions and solving these pairs of equations. Verte A:, Verte B: 5, Verte C:, = 5 = = C = (, ) 5 set A = (, ) B = (5, ) STUDY TIP Using different colored pencils to shade the solution of each inequalit in a sstem makes identifing the solution of the sstem of inequalities easier. The region common to ever graph in the sstem is where all shaded regions overlap. This region represents the solution set of the sstem. Figure F.5 Note in Figure F.5 that the vertices of the region are represented b open dots. This means that the vertices are not solutions of the sstem of inequalities. Now tr Eercise 7.

4 For the triangular region shown in Figure F.5, each point of intersection of a pair of boundar lines corresponds to a verte. With more complicated regions, two border lines can sometimes intersect at a point that is not a verte of the region, as shown in Figure F.. To keep track of which points of intersection are actuall vertices of the region, ou should sketch the region and refer to our sketch as ou find each point of intersection. Appendi F. Solving Sstems of Inequalities A09 Not a verte Figure F. Eample 5 Solving a Sstem of Inequalities Sketch the region containing all points that satisf the sstem of inequalities. Inequalit Inequalit As shown in Figure F.7, the points that satisf the inequalit are the points ling above (or on) the parabola given b. Parabola The points that satisf the inequalit are the points ling below (or on) the line given b. Line To find the points of intersection of the parabola and the line, solve the sstem of corresponding equations. Using the method of substitution, ou can find the solutions to be, 0 and,. So, the region containing all points that satisf the sstem is indicated b the purple shaded region in Figure F.7. Now tr Eercise 55. = + = (, ) (, 0) Figure F.7

5 A0 Appendi F Sstems of Inequalities When solving a sstem of inequalities, ou should be aware that the sstem might have no solution, or it might be represented b an unbounded region in the plane. These two possibilities are shown in Eamples and 7. Eample A Sstem with No Sketch the solution set of the sstem of inequalities. > < Inequalit Inequalit From the wa the sstem is written, it is clear that the sstem has no solution, because the quantit cannot be both less than and greater than. Graphicall, the inequalit > is represented b the half-plane ling above the line, and the inequalit < is represented b the half-plane ling below the line, as shown in Figure F.8. These two half-planes have no points in common. So the sstem of inequalities has no solution. + = + = Figure F.8 No Now tr Eercise 5. STUDY TIP Remember that a solid line represents points on the boundar of a region that are solutions to the sstem of inequalities and a dashed line represents points on the boundar of a region that are not solutions. An unbounded region of a graph etending infinitel in the plane should not be bounded b a solid or dashed line, as shown in Figure F.9. Eample 7 An Unbounded Set Sketch the solution set of the sstem of inequalities. < Inequalit > Inequalit The graph of the inequalit < is the half-plane that lies below the line, as shown in Figure F.9. The graph of the inequalit > is the half-plane that lies above the line. The intersection of these two halfplanes is an infinite wedge that has a verte at, 0. This unbounded region represents the solution set. Now tr Eercise 5. + = + = (, 0) Figure F.9 Unbounded Region

6 Appendi F. Solving Sstems of Inequalities A Applications The net eample discusses two concepts that economists call consumer surplus and producer surplus. As shown in Figure F.0, the point of equilibrium is defined b the price p and the number of units that satisf both the demand and suppl equations. Consumer surplus is defined as the area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, and to the right of the p-ais. Similarl, the producer surplus is defined as the area of the region that lies above the suppl curve, below the horizontal line passing through the equilibrium point, and to the right of the p-ais. The consumer surplus is a measure of the amount that consumers would have been willing to pa above what the actuall paid, whereas the producer surplus is a measure of the amount that producers would have been willing to receive below what the actuall received. Eample 8 Consumer Surplus and Producer Surplus The demand and suppl functions for a new tpe of calculator are given b p p Demand equation Suppl equation where p is the price (in dollars) and represents the number of units. Find the consumer surplus and producer surplus for these two equations. Price p Demand curve Suppl curve Figure F.0 Consumer surplus Equilibrium point Producer surplus Number of units Begin b finding the point of equilibrium b setting the two equations equal to each other and solving for ,000,000 Set equations equal to each other. Combine like terms. Solve for. So, the solution is,000,000, which corresponds to an equilibrium price of p $0. So, the consumer surplus and producer surplus are the areas of the following triangular regions. Consumer Surplus Producer Surplus p p 0 p p In Figure F., ou can see that the consumer and producer surpluses are defined as the areas of the shaded triangles. Consumer surplus (base)(height),000,0000 $5,000,000 Price per unit (in dollars) p Figure F. Suppl vs. Demand p = Producer surplus Consumer surplus p = 0 p = ,000,000,000,000 Number of units Producer surplus (base)(height),000,0000 $90,000,000 Now tr Eercise 75.

7 A Appendi F Sstems of Inequalities Eample 9 Nutrition The minimum dail requirements from the liquid portion of a diet are 00 calories, units of vitamin A, and 90 units of vitamin C. A cup of dietar drink X provides 0 calories, units of vitamin A, and 0 units of vitamin C. A cup of dietar drink Y provides 0 calories, units of vitamin A, and 0 units of vitamin C. Set up a sstem of linear inequalities that describes how man cups of each drink should be consumed each da to meet the minimum dail requirements for calories and vitamins. Begin b letting and represent the following. number of cups of dietar drink X number of cups of dietar drink Y To meet the minimum dail requirements, the following inequalities must be satisfied Calories Vitamin A Vitamin C The last two inequalities are included because and cannot be negative. The graph of this sstem of inequalities is shown in Figure F.. (More is said about this application in Eample in Section F..) STUDY TIP When using a sstem of inequalities to represent a real-life application in which the variables cannot be negative, remember to include inequalities for this constraint. For instance, in Eample 9, and cannot be negative, so the inequalities 0 and 0 must be included in the sstem. Liquid Portion of a Diet 8 Cups of drink Y (0, ) (5, 5) (, ) (, ) (8, ) (9, 0) 8 0 Cups of drink X Figure F. From the graph, ou can see that two solutions (other than the vertices) that will meet the minimum dail requirements for calories and vitamins are 5, 5 and 8,. There are man other solutions. Now tr Eercise 8.

8 Appendi F. Solving Sstems of Inequalities A F. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. An ordered pair a, b is a of an inequalit in and if the inequalit is true when a and b are substituted for and, respectivel.. The of an inequalit is the collection of all solutions of the inequalit.. The graph of a inequalit is a half-plane ling on one side of the line a b c.. The of is defined b the price p and the number of units that satisf both the demand and suppl equations. In Eercises 8, match the inequalit with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) (c) (e) (b) (d) (f). < < > 9 > In Eercises 9 8, sketch the graph of the inequalit. 9. < 0... < < 0. 5 > > > 0. > < 9 > 9 (g) (h) In Eercises 9 0, use a graphing utilit to graph the inequalit. Use the shade feature to shade the region representing the solution < > > 7. < ln 8. ln 5 9. > 0.

9 A Appendi F Sstems of Inequalities In Eercises, write an inequalit for the shaded region shown in the graph < >.. 0 < < > e 0.. In Eercises 5 and, determine whether each ordered pair is a solution of the sstem of inequalities. 5. (a) 0, (b), 5 < (c) 8, (d), < 7. (a), 7 (b) 5, 0 (c), 0 (d), 8 5 In Eercises 7, sketch the graph of the solution of the sstem of inequalities < > 0 0 > > 5 > 5 < > < 5 > > 0 5. < 5. > < > < > 9 > 0 < In Eercises 5 7, find a set of inequalities to describe the region Rectangle: Vertices at,, 5,, 7. Parallelogram: Vertices at 0, 0, 7. Triangle: Vertices at 0, 0, 5, 0,, 7. Triangle: Vertices at, 0,, 0, 0, Suppl and Demand In Eercises 75 78, (a) graph the sstem representing the consumer surplus and producer surplus for the suppl and demand equations, and shade the region representing the solution of the sstem, and (b) find the consumer surplus and the producer surplus. Demand Suppl 75. p p 0.5 5, 7,, 7, 0,,, 5,

10 Demand 7. p p p Suppl p 5 0. p p Appendi F. Solving Sstems of Inequalities A5 (b) Sketch the graph of the sstem in part (a). 8. Graphical Reasoning Two concentric circles have radii of and meters, where > (see figure). The area between the boundaries of the circles must be at least 0 square meters. In Eercises 79 8, (a) find a sstem of inequalities that models the problem and (b) graph the sstem, shading the region that represents the solution of the sstem. 79. Investment Analsis A person plans to invest some or all of $0,000 in two different interest-bearing accounts. Each account is to contain at least $7500, and one account should have at least twice the amount that is in the other account. 80. Ticket Sales For a summer concert event, one tpe of ticket costs $0 and another costs $5. The promoter of the concert must sell at least 0,000 tickets, including at least 0,000 of the $0 tickets and at least 5000 of the $5 tickets, and the gross receipts must total at least $00,000 in order for the concert to be held. 8. Nutrition A dietitian is asked to design a special dietar supplement using two different foods. The minimum dail requirements of the new supplement are 80 units of calcium, 0 units of iron, and 80 units of vitamin B. Each ounce of food X contains 0 units of calcium, 5 units of iron, and 0 units of vitamin B. Each ounce of food Y contains 0 units of calcium, 0 units of iron, and 0 units of vitamin B. 8. Inventor A store sells two models of computers. Because of the demand, the store stocks at least twice as man units of model A as units of model B. The costs to the store for models A and B are $800 and $00, respectivel. The management does not want more than $0,000 in computer inventor at an one time, and it wants at least four model A computers and two model B computers in inventor at all times. 8. Construction You design an eercise facilit that has an indoor running track with an eercise floor inside the track (see figure). The track must be at least 5 meters long, and the eercise floor must have an area of at least 500 square meters. Eercise floor (a) Find a sstem of inequalities describing the requirements of the facilit. (a) Find a sstem of inequalities describing the constraints on the circles. (b) Graph the inequalit in part (a). (c) Identif the graph of the line in relation to the boundar of the inequalit. Eplain its meaning in the contet of the problem. Snthesis True or False? In Eercises 85 and 8, determine whether the statement is true or false. Justif our answer. 85. The area of the figure defined b the sstem below is 99 square units. 8. The graph below shows the solution of the sstem 5 9 > Think About It After graphing the boundar of an inequalit in and, how do ou decide on which side of the boundar the solution set of the inequalit lies? 88. Writing Describe the difference between the solution set of a sstem of equations and the solution set of a sstem of inequalities. 0 8

11 A Appendi F Sstems of Inequalities F. Linear Programming Linear Programming: A Graphical Approach Man applications in business and economics involve a process called optimization, in which ou are asked to find the minimum or maimum value of a quantit. In this section, ou will stud an optimization strateg called linear programming. A two-dimensional linear programming problem consists of a linear objective function and a sstem of linear inequalities called constraints. The objective function gives the quantit that is to be maimized (or minimized), and the constraints determine the set of feasible solutions. For eample, suppose ou are asked to maimize the value of z a b Objective function subject to a set of constraints that determines the region in Figure F.. Because ever point in the shaded region satisfies each constraint, it is not clear how ou should find the point that ields a maimum value of z. Fortunatel, it can be shown that if there is an optimal solution, it must occur at one of the vertices. So, ou can find the maimum value of z b testing z at each of the vertices. Optimal of a Linear Programming Problem If a linear programming problem has a solution, it must occur at a verte of the set of feasible solutions. If there is more than one solution, at least one of them must occur at such a verte. In either case, the value of the objective function is unique. What ou should learn Solve linear programming problems. Use linear programming to model and solve real-life problems. Wh ou should learn it Linear programming is a powerful tool used in business and industr to manage resources effectivel in order to maimize profits or minimize costs. For instance, Eercise on page A shows how to use linear programming to analze the profitabilit of two models of snowboards. Feasible solutions Figure F. Here are some guidelines for solving a linear programming problem in two variables in which an objective function is to be maimized or minimized. Solving a Linear Programming Problem. Sketch the region corresponding to the sstem of constraints. (The points inside or on the boundar of the region are feasible solutions.). Find the vertices of the region.. Test the objective function at each of the vertices and select the values of the variables that optimize the objective function. For a bounded region, both a minimum and a maimum value will eist. (For an unbounded region, if an optimal solution eists, it will occur at a verte.)

12 Appendi F. Linear Programming A7 Eample Solving a Linear Programming Problem Find the maimum value of z subject to the following constraints. 0 0 Objective function Constraints The constraints form the region shown in Figure F.. At the four vertices of this region, the objective function has the following values. At 0, 0: At, 0: At, : z z 0 z 8 Maimum value of z At 0, : z 0 So, the maimum value of z is 8, and this value occurs when and. Now tr Eercise. In Eample, tr testing some of the interior points in the region. You will see that the corresponding values of z are less than 8. Here are some eamples. At, : z 5 At, : z At z 9, : To see wh the maimum value of the objective function in Eample must occur at a verte, consider writing the objective function in the form z Famil of lines where z is the -intercept of the objective function. This equation represents a famil of lines, each of slope. Of these infinitel man lines, ou want the one that has the largest z-value while still intersecting the region determined b the constraints. In other words, of all the lines with a slope of, ou want the one that has the largest -intercept and intersects the given region, as shown in Figure F.5. It should be clear that such a line will pass through one (or more) of the vertices of the region. = 0 = (0, 0) (, 0) = 0 Figure F. Figure F.5 (0, ) + = (, ) STUDY TIP Remember that a verte of a region can be found using a sstem of linear equations. The sstem will consist of the equations of the lines passing through the verte.

13 A8 Appendi F Sstems of Inequalities The net eample shows that the same basic procedure can be used to solve a problem in which the objective function is to be minimized. Eample Solving a Linear Programming Problem Find the minimum value of z 5 7 Objective function where 0 and 0, subject to the following constraints Constraints The region bounded b the constraints is shown in Figure F.. B testing the objective function at each verte, ou obtain the following. At 0, : z 50 7 Minimum value of z At 0, : z At, 5: z At, : z At 5, 0: z At, 0: z So, the minimum value of z is, and this value occurs when 0 and. Now tr Eercise 5. 5 (, 5) (0, ) (, ) (0, ) (, 0) (5, 0) 5 Figure F. Eample Solving a Linear Programming Problem Find the maimum value of z 5 7 Objective function where 0 and 0, subject to the following constraints. 5 Constraints 5 7 This linear programming problem is identical to that given in Eample above, ecept that the objective function is maimized instead of minimized. Using the values of z at the vertices shown in Eample, ou can conclude that the maimum value of z is 5, and that this value occurs when and. Now tr Eercise.

14 Appendi F. Linear Programming A9 It is possible for the maimum (or minimum) value in a linear programming problem to occur at two different vertices. For instance, at the vertices of the region shown in Figure F.7, the objective function z has the following values. At 0, 0: z At 0, : z 0 8 At, : z At 5, : z 5 Objective function Maimum value of z Maimum value of z At 5, 0: z In this case, ou can conclude that the objective function has a maimum value (of ) not onl at the vertices, and 5,, but also at an point on the line segment connecting these two vertices, as shown in Figure F.7. Note that b rewriting the objective function as (0, ) (, ) z = for an point along this line segment. (0, 0) (5, 0) Figure F.7 (5, ) 5 z ou can see that its graph has the same slope as the line through the vertices, and 5,. Some linear programming problems have no optimal solutions. This can occur if the region determined b the constraints is unbounded. Eample An Unbounded Region Find the maimum value of z Objective function where 0 and 0, subject to the following constraints. 7 7 Constraints The region determined b the constraints is shown in Figure F.8. For this unbounded region, there is no maimum value of z. To see this, note that the point, 0 lies in the region for all values of. B choosing large values of, ou can obtain values of z 0 that are as large as ou want. So, there is no maimum value of z. For the vertices of the region, the objective function has the following values. So, there is a minimum value of z, z 0, which occurs at the verte,. At, : z At, : z 0 Minimum value of z At, 0: z 0 Now tr Eercise. 5 (, ) (, ) (, 0) 5 Figure F.8

15 A0 Appendi F Sstems of Inequalities Applications Eample 5 shows how linear programming can be used to find the maimum profit in a business application. Eample 5 Optimizing Profit A manufacturer wants to maimize the profit from selling two tpes of boed chocolates. A bo of chocolate covered creams ields a profit of $.50, and a bo of chocolate covered cherries ields a profit of $.00. Market tests and available resources have indicated the following constraints.. The combined production level should not eceed 00 boes per month.. The demand for a bo of chocolate covered cherries is no more than half the demand for a bo of chocolate covered creams.. The production level of a bo of chocolate covered creams is less than or equal to 00 boes plus three times the production level of a bo of chocolate covered cherries. Let be the number of boes of chocolate covered creams and be the number of boes of chocolate covered cherries. The objective function (for the combined profit) is given b P.5. Objective function The three constraints translate into the following linear inequalities Because neither nor can be negative, ou also have the two additional constraints of 0 and 0. Figure F.9 shows the region determined b the constraints. To find the maimum profit, test the value of P at each verte of the region. At 0, 0: P At 800, 00: P Maimum profit At 050, 50: At 00, 0: P P So, the maimum profit is $000, and it occurs when the monthl production consists of 800 boes of chocolate covered creams and 00 boes of chocolate covered cherries. Boes of chocolate covered cherries Figure F.9 (0, 0) (800, 00) (050, 50) (00, 0) Boes of chocolate covered creams Now tr Eercise 5.

16 In Eample 5, suppose the manufacturer improves the production of chocolate covered creams so that a profit of $.50 per bo is obtained. The maimum profit can now be found using the objective function P.5. B testing the values of P at the vertices of the region, ou find that the maimum profit is now $95, which occurs when 050 and 50. Eample Optimizing Cost The minimum dail requirements from the liquid portion of a diet are 00 calories, units of vitamin A, and 90 units of vitamin C. A cup of dietar drink X costs $0. and provides 0 calories, units of vitamin A, and 0 units of vitamin C. A cup of dietar drink Y costs $0.5 and provides 0 calories, units of vitamin A, and 0 units of vitamin C. How man cups of each drink should be consumed each da to minimize the cost and still meet the dail requirements? Appendi F. Linear Programming A As in Eample 9 on page A, let be the number of cups of dietar drink X and let be the number of cups of dietar drink Y. For Calories: For Vitamin A: For Vitamin C: Constraints 0 The cost C is given b C Objective function The graph of the region determined b the constraints is shown in Figure F.0. To determine the minimum cost, test C at each verte of the region. At 0, : At, : At, : C C C Minimum value of C At 9, 0: C So, the minimum cost is $0. per da, and this cost occurs when three cups of drink X and two cups of drink Y are consumed each da. Now tr Eercise 7. Cups of drink Y Liquid Portion of a Diet 8 (0, ) (, ) (, ) (9, 0) 8 0 Cups of drink X Figure F.0 TECHNOLOGY TIP You can check the points of the vertices of the constraints b using a graphing utilit to graph the equations that represent the boundaries of the inequalities. Then use the intersect feature to confirm the vertices.

17 A Appendi F Sstems of Inequalities F. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. In the process called, ou are asked to find the minimum or maimum value of a quantit.. The of a linear programming problem gives the quantit that is to be maimized or minimized.. The of a linear programming problem determine the set of. In Eercises, find the minimum and maimum values of the objective function and where the occur, subject to the indicated constraints. (For each eercise, the graph of the region determined b the constraints is provided.). Objective function:. Objective function: z Objective function:. Objective function: z 0 7 See Eercise. See Eercise. 5. Objective function:. Objective function: z z z 7 z Figure for 5 Figure for 7. Objective function: 8. Objective function: z See Eercise 5. See Eercise. 9. Objective function: 0. Objective function: z Objective function:. Objective function: z 5 0 z z z 5 0 See Eercise 9. See Eercise 0. 8

18 Appendi F. Linear Programming A In Eercises, sketch the region determined b the constraints. Then find the minimum and maimum values of the objective function and where the occur, subject to the indicated constraints.. Objective function:. Objective function: z Objective function:. Objective function: z Objective function: 8. Objective function: z See Eercise 5. See Eercise. 9. Objective function: 0. Objective function: z See Eercise 5. See Eercise.. Objective function:. Objective function: z Objective function:. Objective function: z See Eercise. See Eercise. 5. Objective function:. Objective function: z z 7 8 z z z z z z See Eercise. See Eercise. Eploration In Eercises 7 0, perform the following. (a) Graph the region bounded b the following constraints. } } ~ ~ (b) Graph the objective function for the given maimum value of z on the same set of coordinate aes as the graph of the constraints. (c) Use the graph to determine the feasible point or points that ield the maimum. Eplain how ou arrived at our answer. Objective Function 7. z 8. z 5 9. z 0. z Maimum In Eercises, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. The objective function is to be maimized in each case.. Objective function:. Objective function: z 0 0. Objective function:. Objective function: z z z 5 z 0 z 5 z z Optimizing Revenue An accounting firm has 800 hours of staff time and 9 hours of reviewing time available each week. The firm charges $000 for an audit and $00 for a ta return. Each audit requires 00 hours of staff time and 8 hours of review time. Each ta return requires.5 hours of staff time and hours of review time. (a) What numbers of audits and ta returns will ield the maimum revenue? (b) What is the maimum revenue?

19 A Appendi F Sstems of Inequalities. Optimizing Profit A manufacturer produces two models of snowboards. The amounts of time (in hours) required for assembling, painting, and packaging the two models are as follows. Model A Model B Assembling.5 Painting Packaging The total amounts of time available for assembling, painting, and packaging are 000 hours, 500 hours, and 500 hours, respectivel. The profits per unit are $50 for model A and $5 for model B. (a) How man of each model should be produced to maimize profit? (b) What is the maimum profit? 7. Optimizing Cost A farming cooperative mies two brands of cattle feed. Brand X costs $5 per bag and contains two units of nutritional element A, two units of nutritional element B, and two units of nutritional element C. Brand Y costs $0 per bag and contains one unit of nutritional element A, nine units of nutritional element B, and three units of nutritional element C. The minimum requirements for nutritional elements A, B, and C are units, units, and units, respectivel. (a) Find the number of bags of each brand that should be mied to produce a miture having a minimum cost per bag. (b) What is the minimum cost? 8. Optimizing Cost A pet suppl compan mies two brands of dr dog food. Brand X costs $5 per bag and contains eight units of nutritional element A, one unit of nutritional element B, and two units of nutritional element C. Brand Y costs $0 per bag and contains two units of nutritional element A, one unit of nutritional element B, and seven units of nutritional element C. Each bag of mied dog food must contain at least units, 5 units, and 0 units of nutritional elements A, B, and C, respectivel. (a) Find the numbers of bags of brands X and Y that should be mied to produce a miture meeting the minimum nutritional requirements and having a minimum cost per bag. (b) What is the minimum cost? Snthesis True or False? In Eercises 9 and 0, determine whether the statement is true or false. Justif our answer. 9. If an objective function has a maimum value at the adjacent vertices, 7 and 8,, ou can conclude that it also has a maimum value at the points.5,.5 and 7.8,.. 0. When solving a linear programming problem, if the objective function has a maimum value at two adjacent vertices, ou can assume that there are an infinite number of points that will produce the maimum value. Think About It In Eercises, find an objective function that has a maimum or minimum value at the indicated verte of the constraint region shown below. (There are man correct answers.). The maimum occurs at verte A.. The maimum occurs at verte B.. The maimum occurs at verte C.. The minimum occurs at verte C. In Eercises 5 and, determine values of t such that the objective function has a maimum value at each indicated verte. 5. Objective function:. Objective function: z t (a) 0, 5 (b), 5 A(0, ) C(5, 0) B(, ) z t 0 0 (a), (b) 0,