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MATHEMATICS (EXTENDED) 0580IGCSE MAY/JUNE 2021LINEARPROGRAMMING150SEKOLAH BUKIT SION - IGCSE MATH REVISION
NOTES: CHAPTER 5 LINEAR PROGRAMMINGLinear Programming is a branch of Mathematics that deals with systems of linear inequalities (calledconstraints) used to findi the maximum or minimum values of the object function.Applications of the Linear Programming are evident in the field of:3 common stages involved in solving Linear Programming problems:Ø Interpret the information given as a system of inequalities and display them graphicallyat least implies less than implies at most implies more than implies Ø Investigate some characteristics of the points in the unshaded solution region (region R)o Utilize “corner points” of RØ Find the maximum/minimum value according to the object function needed.
EXAMPLE:Miguel has 100 to spend on blue and red pens.The cost of each blue pen is 6 while a red pen costs 8.He must buy at least 6 pens of each color, at least 13 pens in total.(a) Represent the information above in inequalities.(b) Show the graph of how he can possibly buy the blue and red pens.(c) If he can sell each blue pen at 8 and each red pen 12, find number of blue and red pens thathe must buy to have the most profit.Working:Let x be the number of blue pens and y be the number of red pens that Miguel buys.If each blue pen is 6, the total cost of x number of blue pens is 6x.If each red pen is 8, the total cost of y number red pens is 8y.If Miguel has a maximum of 100 to spend, the total cost of x blue pens and y red pens is6x 8y 1003x 4y 50 (simplest form by dividing 6x 8y 100 by 2) à Inequality #1If Miguel must buy at least 6 pens of each color:x 6 àInequality #2y 6 àInequality #3Profit: 2x 4yIf he sells each blue pen at 8, his profit for every blue pen is 2.His profit for all blue pens à (2x)If he sells each red pen at 12, his profit for every red pen is 4.His profit for all red pens à (4y)Test Pointà(6,6) à(8,6) à(6,8) à(7,7) àProfit 36 40 44 42His total profit is then (2x 4y).Hence, he must buy 6 blue pens and 8 red pens.151SEKOLAH BUKIT SION - IGCSE MATH REVISION
1. Given the inequalities:x y 11y 3 and y x.Find the point having whole number coordinates and satisfying these inequalities whichgives:(a) the maximum value of x 4yAnswer: [2](b) the minimum value of 3x yAnswer: [2]2. Given:3x 2y 24;x y 12;y ½ x;y 1Find the point having whole number coordinates and satisfying these inequalities whichgives:(a) the maximum value of 2x 3yAnswer: [2](b) the minimum value of x yAnswer: [2]152SEKOLAH BUKIT SION - IGCSE MATH REVISION
3.The region R contain points which satisfy the inequalitiesy !"# 4y 3andx y 6.On the grid, label with the letter R the region which satisfy these inequalities.You must shade the unwanted regions.[3]4. Pablo plants x lemon trees and y orange trees.(a) (i) He plants at least 4 lemon trees.Write down an inequality to show this information.Answer: [1](ii) Pablo plants at least 9 orange trees.Write down an inequality to show this information.Answer: [1](iii) The greatest possible number of trees he can plant is 20.Write down an inequality in x and y to show this information.Answer: [1]153SEKOLAH BUKIT SION - IGCSE MATH REVISION
(b) Lemon trees cost 5 each and orange trees cost 10 each.The maximum Pablo can spend is 170.Write down an inequality in x and y and show that it simplifies to x 2y 34.[2](c) (i) On the grid below, draw four lines to show the four inequalities andshade the unwanted region.[4](ii) Calculate the smallest cost when Pablo buys a total of 20 trees.Answer: [2]154SEKOLAH BUKIT SION - IGCSE MATH REVISION
5. (a) Luke wants to buy x goats and y sheep.(i) He wants to buy at least 5 goats.Write down an inequality in x to represent this condition.Answer: [1](ii) He wants to buy at least 11 sheep.Write down an inequality in y to represent this condition.Answer: [1](iii) He wants to buy at least 20 animals.Write down an inequality x and y to represent this condition.Answer: [1](b) Goat costs 4 and sheep costs 8.The maximum Luke can spend is 160.Write down an inequality in x and show that it simplifies to x 2y 40[2]155SEKOLAH BUKIT SION - IGCSE MATH REVISION
(c) (i) On the grid below, draw four lines to show the four inequalities andshade the unwanted regions.(ii) Work out the maximum number of animals that Luke can buy.156[5][2]SEKOLAH BUKIT SION - IGCSE MATH REVISION
6.Write down the 3 inequalities which define the unshaded region.Answer: [1]Answer: [1]Answer: [4]157SEKOLAH BUKIT SION - IGCSE MATH REVISION
7. Sima sells x biscuits and y cakes.(a) (i) She sells at least 100 biscuits. Write down an inequality in x.Answer: [1](ii) She sells at least 120 cakes. Write down an inequality in y.Answer: [1](iii) She sells a maximum of 300 biscuits and cakes altogether.Write down an inequality in x and y.Answer: [1](iv) Sima makes a profit of 40 cents on each biscuit and 80 cents on each cake.Her total profit is at least 160.Show that x 2y 400.Answer: [2](b) On the grid below, draw four lines to show the four inequalities and shadethe unwanted regions.[4](c) Calculate Sima’s maximum profit.Give your answer in dollars.Answer: [2]158SEKOLAH BUKIT SION - IGCSE MATH REVISION
The region R contain points which satisfy the inequalities y ! " # 4 y 3 and x y 6. On the grid, label with the letter R the region which satisfy these inequalities. You must shade the un
