MATHEMATICS (EXTENDED) 0580 IGCSE MAY/JUNE 2021 LINEAR ...

MATHEMATICS (EXTENDED) 0580 IGCSE MAY/JUNE 2021

LINEAR PROGRAMMING

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SEKOLAH BUKIT SION - IGCSE MATH REVISION

NOTES: CHAPTER 5 LINEAR PROGRAMMING

Linear Programming is a branch of Mathematics that deals with systems of linear inequalities (called constraints) 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:

? Interpret the information given as a system of inequalities and display them graphically

at least implies >

less than implies <

at most implies <

more than implies >

? Investigate some characteristics of the points in the unshaded solution region (region R) o Utilize "corner points" of R

? 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 that

he 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 is 6x + 8y < 100 3x + 4y < 50 (simplest form by dividing 6x + 8y < 100 by 2) ? Inequality #1

If Miguel must buy at least 6 pens of each color: x > 6 ? Inequality #2 y > 6 ? Inequality #3

If he sells each blue pen at $8, his profit for every blue pen is $2. His profit for all blue pens ? $(2x)

If he sells each red pen at $12, his profit for every red pen is $4. His profit for all red pens ? $(4y)

His total profit is then $(2x + 4y).

Hence, he must buy 6 blue pens and 8 red pens.

Profit: 2x + 4y

Test Point? (6,6) ? (8,6) ? (6,8) ? (7,7) ?

Profit $36 $40 $44 $42

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SEKOLAH BUKIT SION - IGCSE MATH REVISION

1. Given the inequalities:

x + y < 11 y > 3 and y < x.

Find the point having whole number coordinates and satisfying these inequalities which gives:

(a) the maximum value of x + 4y

Answer: ................................................ [2] (b) the minimum value of 3x + y

Answer: ................................................ [2]

2. Given:

3x + 2y > 24;

x + y < 12; y < ? x;

y > 1

Find the point having whole number coordinates and satisfying these inequalities which gives:

(a) the maximum value of 2x + 3y

Answer: ................................................ [2] (b) the minimum value of x + y

Answer: ................................................ [2]

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SEKOLAH BUKIT SION - IGCSE MATH REVISION

3.

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 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]

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SEKOLAH BUKIT SION - IGCSE MATH REVISION

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