Inequalities and linear programming

Transcription

1 Inequalities and linear programming. Kick off with CAS. Graphs of linear inequalities. Introduction to linear programming. Applications of linear programming. Review

2 U N C O R R EC TE D PA G E PR O O FS. Kick off with CAS Please refer to the Resources tab in the Prelims section of our ebookplus for a comprehensive step-b-step guide on how to use our CAS technolog. cinequalitiesandlinearprogramming.indd 7// 7:7 PM

3 . Graphs of linear inequalities Linear inequalities When a linear equation is drawn on the Cartesian plane, the plane is divided into two distinct regions or sections. A linear inequalit is a linear equation with the equals sign replaced with an inequalit sign. This sign determines which one of the two regions drawn is the solution to the inequalit. The line which divides the plane into two regions ma or ma not be included in the inequalit, depending on the sign used. Inequalit sign Meaning > Greater than < Less than Greater than or equal to Less than or equal to Graphing linear inequalities When graphing a linear inequalit, the first step is to graph the equivalent linear equation. Once this is done, we need to determine which of the two regions represents our linear inequalit. To do this we can take an point on either side of the line (known as a test point) and substitute the - and -values of this point into the inequalit to determine whether it satisfies the inequalit. Once we ve determined which region of the graph satisfies our inequalit, we need to represent the required region. In this book we leave the required region unshaded; however, ou could chose to shade the required region. Be sure to include a legend with our diagram, as shown, to indicate the required region in this case it is the unshaded region. Finall we need to determine whether or not the line belongs in our inequation. Stle of line A solid line represents: A dashed line represents: Greater than or equal to > Greater than Less than or equal to < Less than = Equal A solid line means that the values on the A dashed line means that the values on line are included in the region. the line are not included in the region. Now we have all of the necessar information needed to graph linear inequalities. Linear inequalities in one variable When graphing a linear inequalit in one variable, the result will either be a vertical line or a horizontal line. Linear inequalities in one variable ma also be displaed on number lines, but in this book we displa them solel on the Cartesian plane. MATHS QUEST GENERAL MATHEMATICS VCE Units and

4 WoRKEd EXAMpLE Sketch the following linear inequalities on separate Cartesian planes, leaving the required regions unshaded. a b > think a For sketching purposes, replace the inequalit sign with an equals sign. Sketch the line =. Determine the stle of the line b looking at the inequalit sign. Select an point not on the line to be our test point. Substitute the - and -values of this point into the inequalit. Check if the statement is true or false, and shade the region that is not required. Add a legend indicating the required region. b For sketching purposes, replace the inequalit sign with an equals sign. Sketch the line =. WritE a b As the inequalit sign is, the line will be solid (meaning values on the line are included). Test point: (, ), so in this case the statement is true and the test point (, ) lies in the required region. So we shade the opposing region, in other words the region opposite to where the test point lies. (, ) Determine the stle of the line b looking at the inequalit sign. Select an point not on the line to be our test point. Substitute the - and -values of this point into the inequalit. As the inequalit sign is >, the line will be dashed (meaning values on the line are not included). Test point: (, ) > > Topic INEQUALITIES ANd LINEAR programming

5 Check if the statement is true or false, and shade the region that is not required. Add a legend indicating the required region. WoRKEd EXAMpLE >, so in this case the statement is true and the test point (, ) lies in the required region. So we shade the opposing region, in other words the region opposite to where the test point lies. Transposing linear inequalities In some situations we need to transpose (rearrange) the inequalit to make or the subject before we can sketch it. When dividing both sides of an inequalit b a negative number, the direction of the sign of the inequalit changes to its opposite direction. For eample, if we are dividing both sides of the linear inequalit < 7 b, then the result is > 7. Note that the less than sign has become a greater than sign. Sketch the linear inequalit + < on a Cartesian plane, leaving the required region unshaded. think Transpose (rearrange) the inequalit so is the subject. For sketching purposes, replace the inequalit sign with an equals sign. Sketch the line =. WritE + < + < < > 7 7 (, ) MATHS QUEST GENERAL MATHEMATICS VCE Units and

6 Determine the stle of the line b looking at the inequalit sign. Select an point not on the line to be our test point. Substitute the - and -values of this point into the inequalit. As the inequalit sign is >, the line will be dashed (meaning the values on the line are not included). Test point: (, ) > > Check if the statement is true or false, and shade the region which is not required. Add a legend indicating the required region. Linear inequalities in two variables Linear inequalities in two variables work in much the same wa as linear inequalities in one variable; however, the must be shown on the Cartesian plane. For eample > + is shown in the diagram below., so in this case the statement is false and the test point (, ) does not lie in the required region. So we shade this region, in other words the region where the test point lies. (, ) 7 7 Notice that the line is dashed to indicate that it does not appear in the required region. Topic Inequalities and linear programming

7 WoRKEd EXAMpLE Sketch the following linear inequalities on separate Cartesian planes, leaving the required regions unshaded. a < + b > think a For sketching purposes replace the inequalit sign with an equals sign. Sketch the line = + using the -intercept and gradient method (-intercept =, gradient = ). Determine the stle of the line b looking at the inequalit sign. Select an point not on the line as our test point. Substitute the - and -values of this point into the inequalit. Check if the statement is true or false, and shade the region that is not required. Add a legend indicating the required region. b For sketching purposes replace the inequalit sign with an equals sign. Sketch the line = using the -intercept and -intercept method. WritE a As the inequalit sign is <, the line will be dashed (meaning the line is not included). Test point: (, ) < + < + < <, so in this case the statement is true and the test point (, ) lies in the required region. So we shade the opposing region, in other words the region opposite to where the test point lies. (, ) b To find the -intercept, = : = = = = The -intercept is at (, ). To find the -intercept, = : = = = = MATHS QUEST GENERAL MATHEMATICS VCE Units and

8 Determine the stle of the line b looking at the inequalit sign. Select an point not on the line to be our test point. Substitute the - and -values of this point into the inequalit. Check if the statement is true or false, and shade the region that is not required. The -intercept is at (, ). As the inequalit sign is <, the line will be dashed (meaning the line is not included). Test point: (, ) > > > 7 > 7 >, so in this case the statement is true and the test point (, ) lies in the required region. So we shade the opposing region, in other words the region opposite to where the test point lies. (, ) Add a legend indicating the required region. Topic Inequalities and linear programming 7

9 Eercise. PRactise Consolidate Graphs of linear inequalities WE Sketch the following linear inequalities on separate Cartesian planes, leaving the required regions unshaded. a b > 7 Write the linear inequalit for each of the following graphs. a b WE Sketch the linear inequalit + < on a Cartesian plane, leaving the required region unshaded. Sketch the linear inequalit + < 7 on a Cartesian plane, leaving the required region unshaded. WE Sketch the following linear inequalities on separate Cartesian planes, leaving the required regions unshaded. a < + b > Complete the linear inequalit for each of the following graphs b placing the correct sign in the bo. a + b + 7 Sketch the following linear inequalities on separate Cartesian planes, leaving the required regions unshaded. a > b < c d Sketch the following linear inequalities on separate Cartesian planes, leaving the required regions unshaded. a + > b < + 9 c + + d + + MATHS QUEST GENERAL MATHEMATICS VCE Units and

10 9 When sketching linear inequalities, which of the following is represented b a dashed line? A + B + C 7 + D > + E + When sketching linear inequalities, which of the following is represented b a solid line? A + > 7 B < + C 7 9 D + < E > + Which of the following is not a suitable test point for the linear inequalit < +? A (, ) B (, ) C (, ) D (, ) E (, ) Which of the following is a suitable test point for the linear inequalit > +? A (, ) B (, ) C (, 9) D (, ) E (, ) Which of the following linear inequalities has been incorrectl sketched? A > + B > C + > 9 D < 7 Topic Inequalities and linear programming 9

11 E 7 + > 7 Sketch the following linear inequalities. a < + b + Which linear inequalit represents the following graph? A > + 7. B < + 7. C + > D + < E > Which linear inequalit best represents the following graph? A + > B > C + < D > E < MATHS QUEST GENERAL MATHEMATICS VCE Units and

12 Master 7 State the inequalit that defines the following graphs. a b State the inequalit that defines the following graphs. a 7 b Introduction to linear programming Simultaneous linear inequalities If we want to solve more than one linear inequalit simultaneousl, we can do so b graphing the solutions for all of the linear inequalities on the same Cartesian plane and finding the intersection of the required regions. Keeping the required region unshaded allows us easil identif which region we require, as we can simpl shade all of the regions that don t fit into the solution. However, ou ma find that some CAS sstems shade the required regions, so use test points to ensure that ou have the correct region. Alwas remember to include a legend with our graphs. Topic Inequalities and linear programming

13 WoRKEd EXAMpLE Find the solution to the following simultaneous linear inequalities, leaving the required region unshaded think WritE Sketch the linear inequalities individuall, To find the -intercept, = : starting with the first inequalit ( + ). + = For sketching purposes, remember to replace the + = inequalit sign with an equals sign. = Sketch the graph of + = using the =. -intercept and -intercept method. The -intercept is at (., ). To find the -intercept, = : + = + = = = The -intercept is at (, ) Determine the stle of the line b looking at the inequalit sign. Select an point not on the line as our test point. Substitute the - and -values of this point into the inequalit. Check if the statement is true or false, and shade the region that is not required. As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) + +, so in this case the statement is true and the test point (, ) lies in the required region. So we shade the opposing region, in other words the region opposite to where the test point lies. MATHS QUEST GENERAL MATHEMATICS VCE Units and

14 Add a legend indicating the required region. Repeat this process with the second inequalit ( + 9). Sketch + = 9 using the -intercept and -intercept method. 9 7 (, ) 7 When = : + = 9 + = 9 = -intercept = (, ) When = : + = 9 + = 9 = 9 -intercept = (, 9) As the inequalit sign is, the line will be solid (meaning the line is included) Select a test point to find the required region. All required regions have now been found, so the remaining unshaded region is the solution (or feasible) region. Test point: (, ) Topic Inequalities and linear programming

15 In this case, the statement is true and the test point (, ) lies in the required region, so we shade the opposing region. 9 7 (, ) 7 Note: The method to find the solution to three or more simultaneous linear inequalities is eactl the same as the method used to find the solution to two simultaneous linear inequalities. Linear programming Linear programming is a method used to achieve the best outcome in a given situation. It is widel used in man industries, but in particular is used in the business, economics and engineering sectors. In these industries, companies tr to maimise profits while minimising costs, which is where linear programming is useful. Constraints in linear programming The constraints in a linear programming problem are the set of linear inequalities that define the problem. In this topic, the constraints are displaed in the form of inequalities that ou are alread familiar with. Real-life linear programming problems ma have hundreds of constraints; however, the problems we will deal with onl have a small number of constraints. Feasible regions When the constraints of a linear programming problem are all sketched on the same grid, the feasible region is acquired. The feasible region is ever required point that is a possible solution for the problem. MATHS QUEST GENERAL MATHEMATICS VCE Units and

16 WoRKEd EXAMpLE Sketch the feasible region for a linear programming problem with the following constraints. + + > > think a Sketch the linear inequalities individuall, starting with the first inequalit ( + ). Remember, for sketching purposes, to replace the inequalit sign with an equals sign. Sketch the graph of + = using the -intercept and -intercept method. Determine the stle of the line b looking at the inequalit sign. Select an point not on the line to be our test point. Substitute the - and -values of this point into the inequalit. Check if the statement is true or false, and shade the region that is not required. WritE a To find the -intercept, = : + = + = = = The -intercept is at (, ). To find the -intercept, = : + = + = = = =. The -intercept is at (,.). As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) , so in this case the statement is true and the test point (, ) lies in the required region. So we shade the opposing region, in other words the region opposite to where the test point lies. Topic INEQUALITIES ANd LINEAR programming

17 Add a legend indicating the required region. (, ) Repeat this process with the second inequalit ( + ). Sketch + = using the -intercept and -intercept method. When = : + = + = = =. -intercept = (., ) When = : + = + = = = -intercept = (, ) As the inequalit sign is <, the line will be a dashed line (meaning the line is not included). 7 Select a test point to find the required region. Test point: (, ) > > + > 9 > MATHS QUEST GENERAL MATHEMATICS VCE Units and

18 Repeat this process with the third inequalit ( > ). Sketch =. 9 Select a test point to find the required region. In this case the statement is true and the test point (, ) lies in the required region, so we shade the opposing region. Place > on the Cartesian plane (use a dashed line due to the > sign). (, ) Test point: (, ) > > The test point (, ) lies in the required region, so we shade the opposing region. (, ) (, ) Topic Inequalities and linear programming 7

19 Repeat this process with the fourth inequalit ( > ). Sketch =. Place > on the Cartesian plane. Use a dashed line due to the > sign. Select a test point to find the required region. All required regions have now been found, so the remaining unshaded region is the feasible region. Test point: (, ) > > The test point (, ) does not lie in the required region, so we shade this region. (, ) identifying the constraints in a linear ProGraMMinG ProBlEM For man linear programming problems, ou won t be given the constraints as linear inequalities, and instead will need to identif them from the tet of the problem. To solve these tpes of problems, first define the variables using appropriate pronumerals, and then identif the ke bits of information from the question necessar to write the constraints as linear inequalities. WoRKEd EXAMpLE The Cake Compan makes two different tpes of cakes: a lemon sponge cake and a Black Forest cake. In order to meet demand, the Cake Compan makes at least batches of lemon sponge cakes a week and at least batches of Black Forest cakes a week. MATHS QUEST GENERAL MATHEMATICS VCE Units and

20 Ever batch consists of cakes, with a batch of lemon sponge cakes taking hours to be made and a batch of Black THINK a Define the variables. Forest cakes taking hours. The equipment used to make the cakes can be used for a maimum of hours a week. a Write the constraints of the problem as linear inequalities. b Sketch the solution to the problem (the feasible region). Write the number of sponge cake batches as a constraint. Write the number of Black Forest cake batches as a constraint. Write the number of sponge and Black Forest cake batches that can be made in the given time as a constraint. b Sketch the first constraint (s ) on a Cartesian plane. Add a legend for the required region. WRITE a Let s = number of sponge cake batches Let b = number of Black Forest cake batches s b.s + b b Test point: (, ) s The test point does not lie in the required region. 7 Black forest cakes 7 9 Sponge cakes Sketch the second constraint (b ) on the same Cartesian plane. Test point: (, ) b The test point does not lie in the required region. Topic Inequalities and linear programming 9

21 Sketch the third constraint (.s + b ) on the same Cartesian plane. The unshaded region is the solution to the problem. Black forest cakes 7 Sponge cakes When s = :.s + b. + b b b 7 Intercept at (, 7).When b = :.s + b.s +.s s 9 Intercept at (, 9).Test point: (, ).s + b The test point lies in the required region. 7 Sponge cakes Black forest cakes The unshaded region in the graph above shows the range of batches that the Cake Compan could make each week. The objective function The objective function is a function of the variables in a linear programming problem (e.g. cost and time). If we can find the maimum or minimum value of the function within the required region, that is, within the possible solutions to the problem, then we have found the optimal solution to the problem. MATHS QUEST GENERAL MATHEMATICS VCE Units and

22 WoRKEd EXAMpLE 7 think a Define the variables. a Emma owns a hobb store, and she makes a profit of $ for ever model car and $ for ever model plane she sells. Write an equation to find her maimum profit (the objective function). b Domenic owns a fast food outlet with his two best-selling products being chips and onion rings. He bus kg bags of chips for $ and kg bags of onion rings for $.. Write an equation to find his minimum cost (the objective function). c A stationer manufacturer makes two tpes of products: rulers and erasers. It costs the manufacturer $. to make the rulers and $. to make the erasers. The manufacturer sells its products to the distributors who bu the rulers for $. and the erasers for $.. Write an equation to find the manufacturer s minimum cost and maimum profit (two objective functions) Determine what is to be maimised or minimised. WritE a Let c = the number of model cars sold Let p = the number of model planes sold Our objective is to maimise the profit. Write the objective function. The objective function is: Profit = c + p b Define the variables. Determine what is to be maimised or minimised. b Let c = the number of kg bags of chips purchased Let r = the number of kg bags of onion rings purchased Our objective is to minimise the cost. Write the objective function. The objective function is: Cost = c +.r c Define the variables. Determine what is to be maimised or minimised. Write the objective functions. To find the profit we need to subtract the costs from the selling price. c Let r = the number of rulers manufactured Let e = the number of erasers manufactured Our two objectives are to minimise the cost and maimise the profit. Profit of ruler =.. =. Profit of eraser =.. =. The objective functions are: Cost =.r +.e Profit =.r +.e Topic INEQUALITIES ANd LINEAR programming

23 Eercise. PRactise Introduction to linear programming WE Find the solution to the following simultaneous linear inequalities, leaving the required region unshaded. + + > Find the solution to the following simultaneous linear inequalities, leaving the required region unshaded. + < 9 > WE Sketch the feasible region for a linear programming problem with the following constraints. + > > > Sketch the feasible region for a linear programming problem with the following constraints. + < 9 > > WE The Biscuit Compan makes two different tpes of biscuits: chocolate cookies and plain biscuits. In order to meet demand, the Biscuit Compan makes at least batches of chocolate cookies a week and at least batches of plain biscuits a week. Each batch consists of individual biscuits. One batch of chocolate cookies takes hour to be made, and one batch of plain biscuits takes an hour. The equipment used to make the biscuits can be used for a maimum of hours a week. a Write the constraints of the problem as linear inequalities. b Sketch the solution to the problem (the feasible region). The Trinket Compan makes two different tpes of trinkets: necklaces and bracelets. In order to meet demand, each week the Trinket Compan makes at least boes of necklaces and at least boes of bracelets. A bo of necklaces takes hours to be made, while a bo of bracelets takes. hours to be made. Each bo contains items. The equipment used to make the trinkets can be used for a maimum of hours a week. a Write the constraints of the problem as linear inequalities. b Sketch the solution to the problem (the feasible region). 7 WE7 Samantha decides to sell items at the countr fair. She makes a profit of $ for ever pair of shoes she sells and $ for ever hat she sells. Write an equation to find her maimum profit (the objective function). MATHS QUEST GENERAL MATHEMATICS VCE Units and

24 Consolidate Morris creates tables and chairs. It costs Morris $. to make a chair and $. to make a table. He sells these items to distributors, who bu the tables for $7. and chairs for $.. Write an equation to help find Morris s minimum cost and maimum profit (two objective functions). 9 (, ) is a feasible solution for which of the following linear inequalities? A + < B < C 7 > D + > E + > (, ) is not a feasible solution for which of the following linear inequalities? A B + > C + > D + < E Which of the following groups of constraints represent the feasible region shown? A > + 9 D > + 9 B < + 9 E < + 9 C < + 9 Which of the following graphs represents the feasible region of the listed constraints? > > + > + 7 Topic Inequalities and linear programming

25 A B MATHS QUEST GENERAL MATHEMATICS VCE Units and

26 C D Topic Inequalities and linear programming

27 E Identif the constraints that represent the following feasible region MATHS QUEST GENERAL MATHEMATICS VCE Units and

28 Master Sketch the feasible regions of the following sets of constraints. a > + + b > + + Rocco is a vet who specialises in cats and dogs onl. On an given da, Rocco can have a maimum of appointments. He is booked for appointments to see at least cats and at least dogs each da. a Determine the constraints in the situation. b Sketch the feasible region for this problem. Write the objective function for the following situations. a Terri sells items of clothing and shoes. She makes a profit of $ for ever piece of clothing she sells and $ for ever pair of shoes she sells. b Emil bus boes of oranges at a cost of $. and boes of avocados at a cost of $. for her fruit shop. c A manufacturing compan makes light globes. Small light globes sell for $. but cost $. to make; large light globes sell for $. but cost $. to make. (two objective functions) 7 A service station sells regular petrol and ethanol blended petrol. Each da the service station sells at least 9 litres of regular petrol and at least litres of ethanol blended petrol. In total, a maimum of litres of petrol is sold on an given da. Let R = the number of litres of regular petrol sold and E = the number of litres of ethanol blended petrol sold. a Identif all of the constraints related to this problem. b Sketch the feasible region for this problem. Dan is a doctor who specialises in knee surger. On an given da, Dan can perform arthroscopies or knee reconstructions. He can perform a maimum of surgeries a week. He is booked weekl to perform at least arthroscopies and at least 7 knee reconstructions. Let A = the number of arthroscopies Dan performs and R = the number of knee reconstructions Dan performs. a Identif all of the constraints related to this problem. b Sketch the feasible region for this problem. 9 Helen bus and sells second-hand fridges and televisions. She bus fridges at a cost of $ and then sells them for $. She bus televisions at a cost of $ and then sells them for $. Let F = the number of fridges sold and T = the number of televisions sold. a Write the objective function for the maimum profit Helen makes. b Helen bus at least fridges and televisions in a ear, with a maimum of items bought in total. Sketch the feasible region for this problem. Topic Inequalities and linear programming 7

29 . WoRKEd EXAMpLE Anna manufactures cutler, specialising in dessert and coffee spoons. It costs Anna $. to make a dessert spoon and $. to make a coffee spoon. She sells these items to distributors, who bu the dessert spoons for $. and the coffee spoons for $.. Let D = the number of dessert spoons made and S = the number of coffee spoons made. a Write the objective function for the maimum profit Anna makes. b Anna manufactures at least twice as man coffee spoons as dessert spoons, and makes less than spoons each week. Sketch the feasible region for this problem. Applications of linear programming The corner point principle After we have found the feasible region for a linear programming problem, all of the points within the feasible region satisf the objective function. The corner point principle states that the maimum or minimum value of the objective function must lie at one of the corners (vertices) of the feasible region. So if we place all of the corner coordinates into the objective function, we can determine the solution to our problem. a Sketch the feasible region of a linear programming problem with the following constraints b Use the corner point principle to determine the maimum and minimum solution of G = +. think a Sketch the inequalities individuall, starting with. For sketching purposes, replace the inequalit sign with an equals sign ( = ). Use the -intercept and -intercept method to sketch the graph, and use a test point to determine the required region. WritE a When = : = = = = The -intercept is (, ). When = : = = = The -intercept is (, ). As the inequalit sign is, the line will be solid (meaning the line is included). MATHS QUEST GENERAL MATHEMATICS VCE Units and

30 Sketch + on the same Cartesian plane. Test point: (, ) is true, so the test point is in the required region and we shade the other region. (, ) When = : + = + = = The -intercept is (, ). When = : + = + = = The -intercept is (, ). As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) + + is false, so the test point is not in the required region and we shade this region. (, ) Topic Inequalities and linear programming 9

31 Sketch + 7 on the same Cartesian plane. When = : + = 7 + = 7 = 7 The -intercept is (7, ). When = : + = 7 + = 7 = 7 The -intercept is (, 7). As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) is true, so the test point is in the required region and we shade the other region. (, ) Sketch on the same Cartesian plane. As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) is true, so the test point is in the required region and we shade the other region. (, ) Sketch on the same Cartesian plane. As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) MATHS QUEST GENERAL MATHEMATICS VCE Units and

32 b Label the vertices of the required region. List the values of the vertices that are easil identifiable. Calculate the coordinates of the remaining points. This is done b solving the simultaneous equations where these points meet. is true, so the test point is in the required region and we shade the other region. (, ) A E B D C A = (, 7) B = (?,?) C = (, ) D = (, ) E = (, ) We need to solve for B: = [] + = 7 [] [] []: = = Substitute = into []: + = 7 = 7 = = B = a, b Topic Inequalities and linear programming

33 Calculate the value of the objective function at each of the corners. At A(, 7): At Ba, b: G = + = + 7 = G = + = + = = At C(, ): At D(, ): G = + G = + = + = + = = At E(, ): G = + = + = State the answer. The maimum value of G is at Ba, b. The minimum value of G is at D(, ). The sliding-line method After we have found the corner points of the feasible region, we can also use the sliding-line method to find the optimal solution(s) to our linear programming problem. To use the sliding-line method ou need to graph the objective function that ou want to maimise (or minimise). As the objective function will not be in the form = m + c, we first need to transpose it into that form. For eample, if the objective function was F = +, we would transpose this into the form = F. This line has a fied gradient, but not a fied -intercept. If we slide this line up (b adjusting the value of F) to meet the last point the line touches in the feasible region, then this point is the maimum value of the function. Similarl if we slide this line down (b adjusting the value of F) to meet the last 7 point the line touches in the feasible region, A(, ) B(, ) then this point is the minimum value of the function. The following graph shows a feasible region with corners C(, ) D(, ) A(, ), B(, ), C(, ) and D(, ). If the objective function was T = +, we would transpose this equation to make the subject: = T. MATHS QUEST GENERAL MATHEMATICS VCE Units and

34 WoRKEd EXAMpLE 9 This equation has a gradient of and a -intercept of T. If we plot this graph for different values of T, we get a series of graphs with the same gradient. To find the maimum value using the objective function, we slide the line = T to the highest verte, in this case A. To find the minimum value using the objective function, we slide the line = T to the lowest verte, in this case D. We can check that these vertices give the maimum and minimum values b calculating T = + at each of the vertices: At A(, ): At B(, ): T = + T = + = = At C(, ): At D(, ): T = + T = + = = a Sketch the feasible region of a linear programming problem with the following constraints b Use the sliding-rule method to determine the maimum and minimum solution of P =. think WritE a Sketch the inequalities a When = : individuall, starting with + = +. For sketching + = purposes, replace the = inequalit sign with an The -intercept is (, ). equals sign ( + = ). When = : Use the -intercept and -intercept method to + = sketch the graph, and use a + = test point to determine the = required region. = 7 The -intercept is (, 7). 7 A(, ) C(, ) D(, ) B(, ) Topic INEQUALITIES ANd LINEAR programming

35 Sketch 9 on the same Cartesian plane. As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) + + is false, so the test point is not in the required region and we shade this region. (, ) When = : = 9 = 9 = The -intercept is (, ). When = : = 9 = 9 = 9 The -intercept is (, 9). As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) MATHS QUEST GENERAL MATHEMATICS VCE Units and

36 Sketch + on the same Cartesian plane. 9 is false, so the test point is not in the required region and we shade this region. When = : + = + = = (, ) The -intercept is (, ). When = : + = + = = The -intercept is (, ). As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) + + is true, so the test point is in the required region and we shade the other region. Topic Inequalities and linear programming

37 Sketch on the same Cartesian plane. (, ) As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) is true, so the test point is in the required region and we shade the other region. (, ) MATHS QUEST GENERAL MATHEMATICS VCE Units and

38 Sketch on the same Cartesian plane. As the inequalit sign is, the line will be solid (meaning the line is included). Test point: (, ) is true, so the test point is in the required region and we shade the other region. b Label the vertices of the required region. List the values of the vertices that are easil identifiable. (, ) C = (?,?) B = (, 9) A = (, 7) D = (, ) E = (, ) A = (, 7) B = (, 9) C = (?,?) D = (, ) E = (, ) Topic Inequalities and linear programming 7

39 Calculate the coordinates of the remaining points. This is done b solving the simultaneous equations where these points meet. Transpose the objective function to make the subject. Choose a value for P and calculate the equation of the line. Draw this line on the graph of the feasible region of the problem. We need to solve for C: = 9 [] + = [] [] []: = =. Substitute =. into []:. + = =. =. B = (.,.) P = P + = =.P +. When P = : =.P +. =. +. =. B = (, 9) A = (, 7) C = (?,?) E = (, ) D = (, ) MATHS QUEST GENERAL MATHEMATICS VCE Units and

40 Slide this line up and down b drawing parallel lines that meet the vertices of the feasible region. C = (?,?) 7 The maimum value of the objective function is at the verte that meets the highest parallel line. The minimum value of the objective function is at the verte that meets the lowest parallel line. Calculate the maimum and minimum values of the objective function. B = (, 9) A = (, 7) E = (, ) D = (, ) The maimum solution is at C(.,.). The minimum solution is at D(, ). At C(.,.): P = =.. = = 9 At D(, ): P = = = = 9 State the answer. The maimum value of P is 9 at C(.,.). The minimum value of P is at D(, ). Solving linear programming problems There are seven steps we need to take to formulate and solve a linear programming problem.. Define the variables.. Find the constraints.. Find the objective function. Topic Inequalities and linear programming 9

41 . Sketch the constraints.. Find the coordinates of the vertices of the feasible region.. Use the corner point principle or sliding-line method. 7. Find the optimal solution. WoRKEd EXAMpLE think Jennifer and Michael s compan sells shirts and jeans to suppliers. From previous eperience, the compan can sell a maimum of items per da. The have a minimum order of shirts and jeans per da. It costs the compan $ to bu a shirt and $ for a pair of jeans, while the sell each shirt for $ and each pair of jeans for $. How man of each item should be sold to make the greatest profit? WritE Define the variables. Number of shirts = Number of jeans = Find the constraints. + Find the objective function. P = + Sketch the constraints to find the feasible region. + Jeans Shirts Test point: (, ) + + is true, so the test point is in the required region and we shade the other region. MATHS QUEST GENERAL MATHEMATICS VCE Units and

42 Jeans Jeans (, ) Shirts Shirts Test point: (, ) is true, so the test point is in the required region and we shade the other region. Jeans Jeans UNCORRECTED PAGE PROOFS (, ) Test point: (, ) Shirts Shirts Topic Inequalities and linear programming

43 Label the vertices of the feasible region and find the coordinates of these vertices. Start with finding the coordinates of A. is false, so the test point is not in the required region and we shade this region. (, ) Shirts A C B Shirts Jeans Jeans At A, the following two lines meet: + So =. Substitute = into the first equation: + = + = = A = (, ) UNCORRECTED PAGE PROOFS Find the coordinates of B. At B, the following two lines meet: + So =. Substitute = into the first equation: + = + = = B = (, ) 7 Find the coordinates of C. At C, the following two lines meet: So C = (, ) MATHS QUEST GENERAL MATHEMATICS VCE Units and

44 Use the corner point principle. P = + A (, ) P = + = 7 B (, ) P = + = 9 C (, ) P = + = 9 Find the optimal solution and write the answer. Eercise. PRactise The optimal solution occurs at point A (, ). Therefore shirts and jeans should be sold to make the maimum profit. Applications of linear programming WE a Sketch the feasible region of a linear programming problem with the following constraints. + 7 b Use the corner point principle to determine the maimum and minimum solution of P = +. a Sketch the feasible region of a linear programming problem with the following constraints. + b Use the corner point principle to determine the maimum and minimum solution of M = 7. WE9 a Sketch the feasible region of a linear programming problem with the following constraints. + b Use the sliding-rule method to determine the maimum and minimum solution of L = +. a Sketch the feasible region of a linear programming problem with the following constraints Topic Inequalities and linear programming

45 Consolidate b Use the sliding-rule method to determine the maimum and minimum solution of S =. WE Gabe and Kim work on their hobb farm. The can have a maimum of animals on their farm due to council regulations. The bu calves at $ and lambs at $, which the then sell when the have matured for $ per cow and $ per sheep. The have a minimum order of cows and sheep from their local animal market seller per month. How man of each animal should be sold to make the greatest profit? Brian s compan sells mobile phones and laptops to suppliers. From previous eperience, the compan can sell a maimum of items a da. The have a minimum order of mobile phones and laptops per da. It costs the compan $ to bu a mobile phone and $ to bu a laptop, and the compan sells each mobile phone for $ and each laptop for $. How man of each item should be sold to make the greatest profit? 7 The Fresh Food Grocer is tring to determine how to maimise the profit the can make from selling apples and pears. The have a limited amount of space, so the can onl have pieces of fruit in total, and the supplier s contract states that the must alwas have pears and apples in stock and no more than pieces of either tpe of fruit. The make a profit of $. for each apple and $. profit for each pear. a Determine how man apples and pears the should have to maimise their profit. b Determine the maimum profit. The following graph shows the feasible region for a linear programming problem. Determine the minimum and maimum values for the objective function S = +. MATHS QUEST GENERAL MATHEMATICS VCE Units and

46 9 Beach Side Resorts are epanding their operations and have bought a new site on which to build chalets and apartments. The want each of their sites to have a minimum of chalets and apartments, and no more than accommodation options in total. Each chalet takes up m in ground space; each apartment takes up m ; and their new site has m of ground space in total. If the can rent the chalets for $ a night and the apartments for $ a night, determine the maimum weekl profit the can make from their new site. A hairdresser offers both quick haircuts and stlised haircuts. The quick haircuts take an average of minutes each and the stlised haircuts take an average of minutes each. The hairdresser likes to do at least quick haircuts and stlised haircuts in an given da, and no more than haircuts in a da. If the hairdresser makes $ on each quick haircut and $ on each stlised haircut, and works for 7 hours in a da: a draw the feasible region that represents this problem b determine how man of each tpe of haircut the hairdresser should do to maimise their dail income c determine their maimum dail income. A new airline compan is tring to determine the laout of the cabins on their new planes. The have two different tpes of tickets: econom and business class. Each econom seat requires. m of cabin space and each business class seat requires. m of cabin space. The airline can make a profit of $ on each econom ticket and $ on each business class seat, although regulations state that the must have at least times more econom seats than business class seats, as well as at least business class seats. If there is a total of m of cabin space for seating: a draw the feasible region that represents this problem b determine how man of each tpe of seat the airline should install to maimise their profit c determine the maimum profit for each flight. Rubio runs an app development compan that creates both simple and comple apps for other companies. His compan can make a maimum of simple apps and comple apps in one week, but cannot make more than apps in total. It takes hours to make each simple app and hours to make each comple app, and the compan can put a maimum of hours per week towards app development. If the compan makes $ profit for each simple app and $ profit for each comple app, how man of each should the aim to make to maimise their weekl profit? Topic Inequalities and linear programming

47 A school is planning an ecursion for all of their students to see the penguins on Phillip Island. The find a compan who can provide two different tpes of coaches for the trip: one that holds passengers and one that holds passengers. The coach compan has of the smaller coaches and larger coaches available. The hire cost $ for a smaller coach and $7 for a larger coach. If there are 7 students and teachers going on the trip, determine the minimum total cost for the coach hire. The feasible region for a linear programming problem is defined b the following constraints: + There are two objective functions for the problem: Objective A =. + 7 Objective B = + a Which objective function has the greater maimum value? b Which objective function has the smaller minimum value? Swish Phone Cases make two different covers for the latest iphone. It takes them minutes to manufacture the parts for case A and minutes to manufacture the parts for case B. There is a total of hours available for manufacturing the cases each week. It takes a further minutes to assemble case A and minutes to assemble case B. There is a total of hours available for assembling each week. Case A retails for $9 and case B retails for $9. a What are the coordinates of the vertices of the feasible region in this problem? b How man of case A and case B should be made each week to maimise profit? c If the prices of case A and case B were swapped, would this affect our answer to part b? Superb Desserts makes two different tpes of chocolate cake, as shown in the following table: Cake Sugar (g) Chocolate (g) Butter (g) Chocolate ripple Death b chocolate The total amount of sugar available to make the cakes is. kg; the total amount of chocolate available is.9 kg; and the total amount of butter is. kg. Each chocolate ripple cake retails for $ and each death b chocolate cake retails for $. Let represent the number of chocolate ripple cakes made and represent the number of death b chocolate cakes made. a Determine the constraints for and. b What is the objective function for the maimum profit? c What is the maimum profit? MATHS QUEST GENERAL MATHEMATICS VCE Units and

48 Master 7 A jeweller compan makes two special pieces of jeweller for Mother s Da. Each piece requires two specialists to work on it; one to shape and set the stone, and one to finish the piece. The first piece of jeweller takes the setter. hours to complete and the finisher.9 hours, while the second piece of jeweller takes the setter. hours to complete and the finisher. hours. Each week the setter can work for up to hours and the finisher can work for up to 7 hours. A profit of $ is made on each of the first pieces of jeweller sold, and a profit of $ is made on each of the second pieces of jeweller sold. a Determine the constraints for the problem. b Draw the feasible region and identif the values of the vertices. c Write the objective function to maimise profit. d Determine how man pieces of each tpe of jeweller should be made each week to maimise profit. e Determine the maimum weekl profit. A pharmac is making two new drugs that are made from the same two compounds. Drug A requires mg of compound P and mg of compound R, while drug B requires 9 mg of compound P and mg of compound R. The pharmac can make a profit of cents for each unit of drug A it sells and cents for each unit of drug B it sells. The compan has. kg of compound P and. kg of compound R to make the drugs, and wants to make at least units of each drug A and units of drug B. a Determine the constraints for the problem. b Draw the feasible region and identif the values of the vertices. c Determine how much of each drug the compan should produce to maimise profit. d Determine the maimum profit. e How much of each compound will remain after making the drugs? Topic Inequalities and linear programming 7

49 ONLINE ONLY. Review the Maths Quest review is available in a customisable format for ou to demonstrate our knowledge of this topic. the review contains: short-answer questions providing ou with the opportunit to demonstrate the skills ou have developed to efficientl answer questions using the most appropriate methods Multiple-choice questions providing ou with the opportunit to practise answering questions using CAS technolog ONLINE ONLY Activities to access ebookplus activities, log on to Interactivities A comprehensive set of relevant interactivities to bring difficult mathematical concepts to life can be found in the Resources section of our ebookplus. Etended-response questions providing ou with the opportunit to practise eam-stle questions. a summar of the ke points covered in this topic is also available as a digital document. REVIEW QUESTIONS Download the Review questions document from the links found in the Resources section of our ebookplus. studon is an interactive and highl visual online tool that helps ou to clearl identif strengths and weaknesses prior to our eams. You can then confidentl target areas of greatest need, enabling ou to achieve our best results. Unit AOS Topic Concept <Topic title to go here> Sit Topic test MATHS Glossar QUEST GENERAL MATHEMATICS VCE Units and

50 Answers Eercise. a b a > b < 7 a b a > + b + 7 Topic Inequalities and linear programming 9

51 7 a b c d a b c d D C 7 MATHS QUEST GENERAL MATHEMATICS VCE Units and

52 B E B a b C A 7 a < + b + a + b > Eercise a c p c +.p Topic Inequalities and linear programming 7

53 b a a n b Plain biscuits b n +.b Bracelets 7 Profit = s + h Cost =.c +.t Profit = 9.c + 9.7t 9 D C C B < > + < + Chocolate cookies Necklaces b a c b 9 7 d c + d Number of dogs a Profit = c + s b Cost = o + a c Cost =.s +.l Profit =.7s +.l Number of cats 7 MATHS QUEST GENERAL MATHEMATICS VCE Units and

54 7 a r 9 b e r + e a A b Ethanol blended petrol sold (L) R 7 A + R Knee reconstructions 9 a Profit = F + T b 9 7 Televisions Regular petrol sold (L) a Profit = d +.7c b Eercise. Arthroscopies Firdges 7 9 Coffee spoons a Dessert spoons b Maimum =, minimum = Topic Inequalities and linear programming 7

55 a a b Maimum =, minimum = 9 9 a b Maimum =, minimum = cows and sheep ($9 ) mobile phones and laptops ($ ) 7 a apples and pears b $ 9 $ a Maimum =, minimum = b quick haircuts and stlised haircuts c $7 b Maimum =, minimum = Stlised haircuts 9 7 Quick haircuts 7 MATHS QUEST GENERAL MATHEMATICS VCE Units and

56 a b econom class seats and business class seats c $ comple apps and simple apps ($) $ a Objective B (9) b Objective B (7) a (, ), (, ), (, ) and (, ) b of each case ($) c Yes, it would then be best to make of case A and of case B ($). a b Profit = + c $9 7 a. +. b Business class seats Second piece of jeweller Econom class seats First piece of jeweller (, ), (, ), (, ) and (, ) c Profit = + d of each tpe of jeweller e $ a b (, ), (, ), (, 7 ) and (, ) c units of Drug A and 7 units of Drug B d $ 7 Drug A e There will be nothing left of either compound. Drug B Topic Inequalities and linear programming 7