ARROGANCE in My Add Maths Module Form V (Version 2010) Topic 21 by NKL 10.1 Region That Satisfies A Linear Inequalities 1. For a straight line ax + by = c, If b > 0 If b < 0 y y ax + by c ax + by = c ax + by c ax + by = c ax + by c ax + by c x x Example 1: y – x = 4 Example 2: x – y = 1 y y y–x 4 y–x =4 x–y 1 2x – y = 1 x–y 1 y–x 4 4 0 x -1 0 x Example 3: 2y + 3x – 1 = 0 Example 4: 6 – 3x = 2y y y 2y + 3x – 1 = 0 6 – 3x = 2y 3 – 3x 2y 2y + 3x – 4 0 3 2 0 x 0 x 2y + 3x – 1 0 6 – 3x 2y Exercise 10.1 Shade the region R on each of the following graphs that satisfies the linear inequalities given. 1) 2y x; y + x 6; y 0 2) y – x 2; x 5;, y 0 y y y–x=2 y+x=6 x=5 2y = x 0 x 0 x 3) x 4; y 2x+1; y + x – 3 0 4) y − x; x 0; 2y + x + 5 0 y x=4 y = 2x + 1 y x 0 2y + x + 5 = 0 x y = −x 0 y+x–3 =0 10.2. Mathematical Statements and Linear Inequalities No. Mathematical Statement Inequality No. Mathematical Statement Inequality 1. y is more than x y>x 7. The maximum value of x is k y k 2. y is less than x y<x 8. The minimum value of x is k xk 3. y is not more than x y x 9. Sum of x and y is not more than k x + y kx 4. y is not less than x y x 10. y is more than x by at least k y xk 5. y is at least k times of x y kx 11. y is not more than k times of x y kx 6. y is at the most k times of x y kx 12. y must exceed x by at least k y xk Exercise 10.2 Write the equalities, beside x 0 and y 0, which describe the following situations. 1. The price of an exercise book is 60 sen and 2. An apple is priced 80 sen and an orange is the price of a pen is RM1.00. A student 60 sen. A housewife has bought x apples wants to buy x exercise books and y pens and y oranges which satisfy the following which satisfy the following conditions. conditions. I. At least 3 pens are bought. I. At least 10 apples are bought. II. total number of exercise books and II. The total number of oranges and apples pens bought must not exceed 10. are not more than 20. III. Amount spent is at the most RM8.00 III. The number of apples exceed two times the number of oranges. IV. Amount spent is at the most RM50. 3. A man has RM90 in coins. He has at least 4. A bakery shop sells two types of bread: Bread RM15 in twenty sen coins and at least RM40 A and Bread B. A piece of Bread A costs in fifty sen coins. The fifty sen coins are two RM2.20 while a piece of Bread B costs times more than the twenty sen coins in RM2.50. The pieces of Bread A and Bread B number. Given the number of twenty sen sold are x and y respectively. The following coins is x and the number of fifty sen coins is condition conditions must be satisfied. y. Write three inequalities that satisfy the I. The sale of Bread A is at most RM1.60. above conditions. II. The total number of bread sold is less than 500. III. The total sale of the types of bread is at least RM400 10.3 Objective Functions To determine the objective or the goal of decision-maker or to make decision that refers to optimum (maximise or minimise) quantity of k and is written in; ax + by = k, where a, b and k are constants. EXERCISE 10.3 Determine the objective function ax + by = k in each of the following situations. 1. The production costs of two kinds of cake, 2. Samad buys x units of hand phones and y A and B are 60 sen and 80 sen per piece units of watches as prizes for a respectively. If a company produces x competition. The costs of a hand phone pieces of cake A and y pieces of cake B, and a watch are RM400 and RM350 form an objective function for the total respectively. Form an objective function cost of production. for the total expenditure. Answer: Answer: 3. The daily wages of a skilled and a non 4. A shop has sold x units of washing skilled worker are RM100 and RM50 machines and y units of refrigerators in a respectively. If a contractor employs x certain month. The profits obtained from skilled workers and y non-skilled each unit of them are RM250 and RM300 workers, form an objective function for respectively. Form an objective function the total wage of the workers. for the total maximum profit. Answer: Answer: 10.4 Solving Problem Related to Linear Programming Linear programming is a technique to determine the optimum (maximum or minimum) of a linear function in two variables under certain constraints or conditions. The possible solutions to the problem are known as feasible solutions. The linear function for which the optimum value is to be determined is known as the objective function. Four steps in solving a linear programming problem. 1. Write the linear inequalities and equations that describe the situations, conditions or constraints of the problem. 2. Draw the graphs of the equations described and shade the region of feasible solutions. 3. Form and draw the objective function. 4. Determine graphically the optimum value of the objective function. EXERCISE 10.4 1. The price of one ream of white paper and that of newsprint paper are RM8 and RM4 respectively. Johan buys x reams of white paper and y reams of newsprint paper based on the following conditions; I. The number of reams of white paper is more than that of newsprint paper. II. Johan buys at least 50 reams of paper. III. Johan has only RM200 to buy all the paper. (a) Write the three inequalities, other than x 0 and y 0, that describe the situations. (b) Using a scale of 1 cm to 5 reams, construct and shade the region of feasible solutions, R that satisfies all the above constraints. (c) By using the graphs in (b) find (i) the minimum number of white paper need to buy if the number of newsprint paper is 5 reams, (ii) the maximum cost Johan has to spend to buy all the paper. 2. A factory has x long buses and y short buses to provide transport for its workers. The factory has 600 workers. The capacities of a long bus and a short bus are 40 and 30 passengers respectively. Given the number of long buses is at the most 12 and the total number of buses is not more than 20. (a) Write down three inequalities, other than x > 0 and y > 0, which satisfy all the above conditions. [3 marks] (b) Using the scale of 1 cm to 2 buses on both axes, construct and shade the region R that satisfies all the above conditions. [3 marks] (c) From the graph, determine the daily minimum and maximum operation costs if the daily operation costs for the long and short buses are RM30 and RM20 respectively. [4 marks] 3. The number of Form I and Form 4 students admitted into a residential school are x and y students respectively. Selection of these students are based on the following criteria: I: The number of Form 4 students exceeds the number of Form I students by not more than 150. II: The number of Form 4 students is at least ¾ of the number of Form I students. III: The total number of Form I students is not more than 450. (a) Write down the three inequalities in x and y which satisfy the above criteria. [3 marks] (b) By using a scale of 2 cm to 50 students on both the x-axis and y-axis, construct and shade the region R which is defined by the above criteria. [3 marks] (c) The monthly fees that each Form I and Form 4 students has to pay are RM25 and RM30respectively. Using the graph in part (b), find (i) the maximum total monthly fee collected from students of these forms, (ii) the maximum and minimum total monthly fees collected from students of these forms given that there are 200 Form 4 students. [4 marks] 4. A tailor takes orders for short pants and trousers only. The time taken to ‘cut’ the materials and then ‘sew’ the materials for each pair of short pants and trousers are as given in the following table. Clothes Cutting Time (minutes) Sewing Time (minutes) Short pants 30 40 Trousers 40 90 The short pants and trousers are made according to the following constraints: I. The minimum time for ‘cutting’ the materials is 9 hours. II. The time for sewing is at least 6 hours 20 minutes. III. The ration of number of pairs of short pants to the number of pairs of trousers is not greater than 3 : 4. In a particular time period, the shop manages to make x pairs of short pants and y pairs of trousers. (a) Write down three inequalities, other than x 0 and y 0, for above constraints. (b) By using a scale of 2 cm to 2 pairs of short pants and 2 cm to 2 pairs of trousers, draw and shade a region R that satisfies all the above constraints. (c) By using the graph in part (b), find (i) the minimum number of pairs of trousers that are made if 4 pairs of short pants are made by the tailor, (ii) the maximum profit made by the tailor if each pair of short pants and trousers net a profit of RM15 and RM9 respectively.