UNIT FOUR LINEAR INEQUALITIES AND LINEAR PROGRAMMING 15 HOURS MATH 521B Revised Jan 30, 01 90
SCO: By the end of grade 11 students will be expected to: A1 relate sets of numbers to solutions of inequalities A7 demonstrate and apply an understanding of discrete and continuous number systems C12 solve linear, simple radical, exponential and absolute value equations or linear inequalities C29 investigate, make and test conjectures about the solutions to equations and inequalities using graphing technology Elaboration- Instructional Strategies/Suggestions 1 - Dimensional Inequalities (2.1) Invite students to investigate only a few of the algebraic problems. Use the TI-83 to re-inforce the students understanding of the topic. Initiate a discussion to allow students to review describing inequalities, both in words and in symbols. They should also investigate the different appearances of discrete and continuous inequality graphs. 2 - Dimensional Inequalities (2.3) Students will not have seen this topic in Junior High so a teacher can t assume it is a review. Spend some time on a few of the basic problems in MathPower 11 p.75 # 1-34 or Math 11 p.348-351. Try to extend the student s experience into the Application Problems where the problems are more real world. Challenge students to find a connection between the three mathematical conditions: <, > and = and the points on the Cartesian Plane. A helpful idea might be to draw a line on the blackboard and ask students to examine the number of different regions (sets of points) formed. Hopefully students discover the 3 regions formed. As well, students may discover that when an equation is graphed, the points that satisfy the = condition are on the line. The points on either side of that line(called the Boundary Line) exist but have not been considered to date. Each of these other two regions will satisfy the inequality conditions > or <. If the inequality is written in slope-intercept form then students may be able to do a few problems and intuitively make a deduction that the > region is above the B. Line(or to the right of it for a vertical line), the < region is below the B. Line (or to the left of it for a vertical line) example x + y > 2 x + y = 2 Once the!x + 2 is entered, cursor over to the left of the Y 1 and press enter until the shaded triangle shown is displayed. This represents > region, while the lower triangle that can be displayed by clicking enter once more represents the < region. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 91
1 Dimensional Inequalities (2.1) Pencil/Paper/Technology Solve algebraically and graphically 2x + 1 > 2 On the TI-83 y = screen enter 2x + 1 2nd test:3:> 2 press zoom 4:Decimal now press Trace and cursor along to do a Boolean test or simply enter various x values and press enter Another possibility would be to look at the Table of Values. 1- Dimensional Inequalities MathPower 11 p63 #25-49 odd Applications MathPower 11 p.63 #51, 54-56, 60 Note to Teachers: When graphing 1-Dimensional Inequalities, the TI-83 use Boolean algebra( If... then statements that are either true or false). If true then y = 1, if false then y = 0. Problem Solving p.67 # 1,2,3,5,7,9,10 2- Dimensional Inequalities (2.3) Journal With the accompanying graph, get the equation of the B. Line and state the inequality that describes the shaded region. Write a real world situation where this might apply. 2- Dimensional Inequalities MathPower 11 p.75 #1-21 odd, 29-33 odd,42 Applications A store makes $100 profit on every pair of downhill skis sold and $60 on every pair of ski boots sold. The store s target is to make at least $600 a day on sales from these 2 items. a) write an inequality that describes the situation b) what are the constraints(restrictions) on the variables c) graph the inequality constraints b = # boots sold b$ 0 s = # skis sold s $ 0 objective quantity 60b + 100s $ 600 Problem Solving p.79 #1,2,7 92
SCO: By the end of grade 11 students will be expected to: C19 graph equations and/or inequalities and analyze graphs both with and without graphing technology C29 investigate, make and test conjectures about the solutions to equations and inequalities using graphing technology Elaboration- Instructional Strategies/Suggestions Systems of Linear Inequalities (2.5) Invite students to explore the inequality regions formed by the B. Lines. Only the region that makes all inequality statements true is shaded. Textbooks typically treat the B. Lines incorrectly (see MathPower 11 p.81, Math 11 p.352) so the students should discuss the issues involved here. The teacher may choose to use the shade feature which yields a neater graph. It also forces the student to deal with domain and range. o Find the intersection set of: ( x, y) 2x + y 4 I x y > 2 Students must be able to read this as: Find the set of ordered pairs(points) that are either above or on the first Boundary Line and below the second Boundary Line. Step 1 Rearrange the statements into y, mx + b form. Î y $! 2x + 4 and Ï y < x! 2 Step 2 Draw both Boundary Lines as broken lines. Step 3 Find the inequality region that is above B. line Î and below B. line Ï. Shade this region. Step 4 Look at whether parts of the Boundary Lines should become solid or not. Since statement Î has an equal sign then the part of that B. line that borders on the shaded region becomes a solid line. Step 5 Since statement Ï does not have an equal sign this B. line remains broken. Step 6 The intersection point of the B. lines is not part of the solution set and is therefore an open circle. q Note to Teachers: This section might be perhaps be best done using pencil and paper. If you choose to let students use graphing calculators then they should be aware of the corrections needed to make the graphical solution correct. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 93
Systems of Linear Inequalities (2.5) Pencil/Paper Which of the following ordered pairs are solutions to the systems of inequalities? x! 2y > 3 3x + 2y # 6 a) (0,0) b) (2,3) c) (0,!3) d) (!4, 0) Pencil/Paper Write a system of inequalities represented by the graph: Systems of Linear Inequalities MathPower 11 p.85 #1-5 odd, 15-45 odd Applications p.86 #47,49,50,60,62 Journal/Communication Write a set of clear instructions that would enable someone to be able to graph a system of inequalities on the TI-83. Communication Without graphing, describe the graphical solution to the system of inequalities below: 2x + y < 4 x! 6y < 18 Technology Solve the following system of inequalities using the TI-83: x! y # 2 x # 3 Note to Teachers To graph a vertical line on the TI-83 2nd draw 4: vertical then cursor to the right or left to position the line properly 94
SCO: By the end of grade 11 students will be expected to: C19 graph equations and/or inequalities and analyze graphs both with and without graphing technology Elaborations - Instructional Strategies/Suggestions Students can also use the following method on the TI-83 to graph inequalities: Graph the solution set of x + y < 3 and x! y > 1 1) convert the inequalities to slope- intercept form: y <!x + 3, y < x!1 2) enter these into the TI-83 on the y = screen 3) to get the < signs, move the cursor to the left of Y 1 and Y 2 and press enter until the correct symbols appear. C29 investigate, make and test conjectures about the solutions to equations and inequalities using graphing technology pressing zoom 6 yields the double shaded region and is the solution. Students should be made aware of the limitations of the calculator in its inability to deal with the Boundary Lines properly. This mistake is mirrored as well in both resources. If the students opt to use the draw feature, the steps are: 1) convert the inequalities to slope-intercept form 2) enter these into the TI-83 on the y = screen 3) set an appropriate window ( here it can be zoom 6 ) 4) press graph then 2nd calc 5:intersect and press enter 3 times to get the intersection point ( here it is (2,1) note: this is necessary for setting the bound in the shade window 5) press 2nd draw 7: shade then enter To shade the rest of the lower region, the student could enter a second row in the shade screen NOTE: The shade feature is for the teachers information and you may not choose to have the students use this method. 95
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Systems of Linear Inequalities Pencil/Paper Discussion A company makes snowmobiles and seadoos. In one day, the assembly line can produce no more than 20 snowmobiles and no more than 30 seadoos. As well, no more than 40 total can be made in any one day. Draw a graph to represent the numbers that can be made in one day and discuss, in your group, the allowed integral solutions. Solution: x = the number of snowmobiles made per day y = the number of seadoos per day Systems of Linear Inequalities Problem Solving p.89 #4-7,10 x # 20 y # 30 x + y # 40 96
SCO: by the end of grade 11 students will be expected to: B4 identify and calculate the maximum and/or minimum values in a linear programming model C27 apply linear programming to find optimal solutions to real world problems C28 express and interpret constraints using inequalities C29 investigate, make and test conjectures about the solutions to equations and inequalities using graphing technology Elaborations - Teaching Strategies/Suggestions Linear Programming (p.91) Let students attempt problems where the constraints and objective function are given such as in p.91green 3 #1(a), 2(a) and 3(a). Perhaps allow students to work in groups and try #1(a). Once all groups have made an attempt to solve the problem, a discussion could ensue where students should be introduced to the following terms: constraints - feasible region - the inequalities the shaded region that satisfies all the inequality constraints objective function - an algebraic function that for students at this level is maximized or minimized at one of the vertices of the feasible region Notes for the teacher: Many students think of mathematics as something that is a finished work and that all the math being done in school was developed by people centuries ago. In fact, mathematics is an alive and exciting subject. One such example is the area of Linear Programming. Its development began during the Second World War. The saying Necessity is the mother of invention applies here. During the war there was a need to coordinate manpower, resources and industry. After the war many countries economies were either devastated or on the brink of collapse, governments and industries put out the challenge to the best mathematical minds for a solution. Linear Programming was the result. One important use of Linear Programming is to maximize profits and minimize costs. Another branch is used for Network Problems. In the context of Linear Programming, the term network is most often associated with the minimumcost network flow problem. A network for this type of problem is viewed as a collection of nodes (locations) and arcs (routes) connecting the nodes. A variety of well-known network problems including: shortest path problems, transportation problems, maximum flow problems and others can be solved with Network Linear Programs that are available( some as free downloads on the internet). One of the major developers of Linear Programming is George Dantzig. During his university days in the 1940's, George came late one day to his statistics class. Seeing a couple of problems on the board and thinking they were for homework, he copied them down. Later, when he had a chance to try them, he found them to be quite difficult. Once he finally had them solved, he turned the solutions in to his professor and apologized for his lateness in completing his assignment. To his surprise, the problems had been previously thought to be unsolvable. The solutions eventually became the basis for his doctoral thesis. His solution became known as the Simplex Method. The essence of this method is moving from one vertex to another along the boundary of the feasible region. This is the method that we will use in this section because inevitably the maximum and minimum points are found among the vertices. A second method of solving Linear Programming problems is the Interior Point Method where points inside the feasible region are investigated. This method has been developed quite recently, in 1984, by Karmarkar. Not every linear programming problem involves finding the maximum or minimum and this is where the second method comes into play. 97
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Linear Programming (p.91) Group Activity/ Technology Use the TI-83 to graph the following systems of inequalities. Write the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the objective function. 1(a) p.91 MathPower 11 Step 1 not necessary here Step 2 Step 3 Constraints Objective Function x$ 0 x + 4y y $ 0 2x + y # 8 x + 2y # 10 Step 4 The first two constraints limit us to look at only the first quadrant and we will take care of these with our window dimensions. Enter the other two constraints into the y = screen Linear Programming MathPower 11 p.91 Green #1(a), 2(a) 3(a) To shade the feasible region we can do it one of three ways: yielding All the opposite regions can be graphed and the solution left blank: or using the shade feature yielding 9 98
SCO: By the end of grade 11 students will be expected to: B4 identify and calculate the maximum and/or minimum values in a linear programming model C27 apply linear programming to find optimal solutions to real world problems C28 express and interpret constraints using inequalities Elaborations - Teaching Strategies/Suggestions Linear Programming (cont d) Linear Programming is widely used everyday in many, many fields including: transportation, energy, telecommunications and manufacturing to name but a few. Many university departments have linear programming divisions turning out specialists that are highly sought after by different levels of government and industry. Challenge students to read MathPower 11 p.92 and, in groups, create a stepby-step method of solving linear programming problems. Note to teachers: The basic steps are: 1) determine what x and y should represent 2) determine the constraints 3) write the objective function 4) graph the constraints to find the feasible region and find the coordinates of the feasible region 5) substitute these coordinates into the objective function to locate the maximum or minimum (whichever is desired) 6) write a conclusion using the coordinates of the maximum or minimum point. In the Worthwhile Tasks are a number of solved problems, some of which are found in MathPower 11 p.92-93. This is an important section and therefore the solutions for most problems are provided for the teacher in order to make class preparation easier. This section shows where all the previous work with linear equations and inequalities was taking the student. C29 investigate, make and test conjectures about the solutions to equations and inequalities using graphing technology Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 99
Linear Programming (cont d) Step 5 and 6 Maximum is at (0,5), where x + 4y = 20 Minimum is at (0,0) where x + 4y = 0 Linear Programming MathPower 11 p.92 #1 P.93 #1, 3-5 #1 p.92 MathPower 11 Step 1 x = # $300 spots y = # $200 spots Step 2 Step 3 Constraints objective function x $ 0 8000x + 6000y (# people y$ 0 hearing the ads) x + y # 10 300x + 200y # 2400 (3x + 2y # 24) Step 4 graph constraints Step 5 The objective function is a maximum of 68,000 at (4,6). Step 6 If four $300 ads and six $200 ads are run, Kim will have the maximum coverage with 68,000 people hearing her ad. Journal Describe a pattern as to the quadrant of the Cartesian Coordinate Plane that is used for these problems. Can you think of a reason as to why this is the case? 100
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Linear Programming (cont d) #2 p.93 MathPower 11 Step 1 x = # hours at hardware store y = # hours at fitness centre Step 2 Step 3 Constraints Objective function x $ 7 15x + 13y earnings to y $ 5 be maximized x + y # 15 Step 4 Graph to draw x = 7 2 nd draw 4:vertical and cursor as close as possible to 7 and press enter Step 5 The maximum occurs at (10,5) and is $215. Step 6 Anna earns a maximum of $215 by working 10 hours at the hardware store and 5 hours at the fitness centre. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 101
Linear Programming (cont d) #3 p93 MathPower 11 Step 1 x = # mogul jackets y = # speed jackets Step 2 Step 3 Constraints Objective function x + y # 20 50x + 80y 100x + 200y # 2800 maximum profit that can x $ 0 be made by spending no y $ 0 more than $2800 Step 4 graphing The vertices and graph are Step 5 The maximum profit of the objective function occurs at (12,8). Step 6 A maximum profit of $1240 happens when Mike sells 12 Mogul jackets and 8 Speed jackets. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 102
Linear Programming (cont d) #4 p.93 MathPower 11 Step 1 x = # regular models y = # deluxe models Step 2 Step 3 Constraints Objective function 2x + 2y # 22 150x + 200y 2x + 5y # 40 maximize profit x $ 0 y $ 0 Step 4 Step 5 The maximum occurs at (5,6) and has a value of $1950. Step 6 A maximum profit of $1950 happens when the company sells 5 regular tables and 6 deluxe tables. Group Project/Communication Design a real world situation where linear programming will be of use in solving the problem. Include sufficient information to allow a person to obtain the constraints and objective function. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 103
Linear Programming (cont d) #5 p.93 MathPower 11 Step 1 x = # classic dishwashers y = # deluxe dishwashers Step 2 Step 3 Constraints Objective function x $ 0 250x + 300y y $ 0 cost function to be x + y $ 100 minimized 50x + 100y $ 6000 Step 4 Step 5 Minimum cost is at (80,20) and has a value of $26000. Step 6 A minimum cost of $26000 is accomplished when 80 classic and 20 deluxe dishwashers. Research Write a few paragraphs on George Dantzig. Try searching the internet to find information on his university days and on his contributions to mathematics. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 104
Linear Programming (cont d) Enrichment A company produces TV s and stereos. They are produced at two different work-stations, with each station involved in making both items. If work-station 1 only made TV s it could produce 40 per day, while if it made only stereos it could make 60 per day. Work-station 2 could produce 50 of either one alone per day. The company makes a profit of $200 on each TV and $150 on each stereo. Assuming the company can sell all the units it can produce, how many units of each should the company produce on a daily basis to maximize its profit. Step 1 x = # TV s produced daily y = # stereos produced daily Problem Solving p.97 #5,9 Extra problems Algebra, Structure & Method Book 2 p. 159 example p.161 #1-5 Worksheet at end of the unit Step 2 Step 3 Constraints Objective function x $ 0 200x + 150y Profit y $ 0 (1/40)x + (1/60)y # 1 (Work-station 1) (1/50)x + (1/50)y # 1 (Work-station 2) Step 4 Step 5 At (20,30) profit reaches a maximum of $8500. Step 6 A maximum profit of $8500 occurs when 20 TV s and 30 stereos are produced daily. 105
Extra Linear Programming Problems 1. A manufacturer makes sneakers and dress shoes. The length of time each type of footwear requires on cutting and stitching machines is shown; Time on cutting machine(minutes) Time on stitching machine(minutes) sneaker 2 2 dress shoe 2 4 Each day the cutting machine runs for only 120 minutes, while the stitching machine is available for 160 minutes. If the profit on a pair of sneakers is $20 and the profit on a pair of shoes is $22, find how many of each type should be made to maximize profits? 2. To manufacture fast-balls and baseballs, a firm uses two machines, A and B. The time required on each machine is shown in the table below. Machine A is available for 9 hours a day and machine B is available for 10 and 2/3 hours each day. If the profit on a fast-ball is $3.20 and the profit on a baseball is $1.20, find how many of each type should be made to maximize profits. Time on Machine A Time on Machine B Baseballs 2 min 1 min Fast-balls 1 min 2 min 3.A toy manufacturer wants to minimize the costs for producing two types of toy cars. Because of a limited supply of materials, no more than 40 racing cars and 60 convertibles can be built each day. There are enough plant workers to make at least 70 cars per day. It costs $12 to manufacture a racing car and $8 to build a convertible. Find how many of each toy should be produced daily to minimize costs. 4. A lumber company manufactures lumber and plywood. In a given week, the total production cannot exceed 800 units, of which 200 units of lumber and 300 units of plywood are pre-sold to their regular customers. The profit on a unit of lumber is $20 and the profit on a unit of plywood is $30. Find the maximum number of units of each that should be manufactured to maximize profits. 5. A toy company makes toys at two different plants A and B. Plant A has materials to make up to 1000 tractors and dolls. Plant B has materials to make up to 800 tractors and cars. Plant A can make 10 tractors and 5 cars per hour. Plant B can make 5 tractors and 15 cars per hour. It costs $300 per hour to operate plant A and $350 per hour to operate plant B. How many hours should each plant be run to minimize costs? 6. A manufacturer makes push-mowers and riding mowers. At least 500 riding mowers and 700 push mowers are needed to meet minimum daily demands. The machinery can produce no more than 1200 riding mowers and 1400 push mowers per day. The combined number that the shipping department can handle is 2300 per day. If riding mowers sell for $2000 and push mowers sell for $400, what is the maximum daily sales the company can expect? 106