Cambridge International Mathematics

EXERCISE 32C

This section is an extension to the syllabus item7.7and is a good example ofmathematical modelling.

In some problems we need to construct our own inequalities. These inequalities are calledconstraints.A

table may be used to help sort out the given information.

Example 6 Self Tutor

Consider the production of 2 -drawer and 5 -drawer filing cabinets. We only have 34 drawers, 8 locks,

and 42 square metres of sheet metal available.

Each 2 -drawer cabinet requires 1 lock and 2 square metres of sheet metal, while each 5 -drawer cabinet

requires 1 lock and 4 m^2 of sheet metal. Letxbe the number of 2 -drawer filing cabinets made andy

be the number of 5 -drawer filing cabinets made. List the constraints connectingxandy.

We put the data in table form:

Type Number of cabinets Locks per cabinet Metal per cabinet

2 -drawer x 1 2 m^2

5 -drawer y 1 4 m^2

x> 0 , y> 0 since we cannot make negative cabinets.

The total number of drawers=2x+5y,so 2 x+5y 634.

The total number of locks=x+y,sox+y 68.

The total number of m^2 of metal=2x+4y,so 2 x+4y 642.

D PROBLEM SOLVING (EXTENSION)

Inequalities (Chapter 32) 645

1 Consider the regionRdefined by: x> 0 , x+2y> 8 and x+y 66.

a Graph the regionRon grid paper.

b How many points inRhave integer coordinates?

c How many of these points obey the rule y¡x=4?

d Find the greatest and least values of 2 x+y for all (x,y) 2 Rwith integer coordinates.

2 Consider the regionRdefined by: x> 0 , y> 0 , x+y 68 and x+3y 612.

a Graph the regionRon grid paper.

b Find the largest value of the following and the corresponding values of integersxandy:

i 2 x+3y ii x+4y iii 3 x+3y.

3aGraph the regionRin which x> 0 , y> 0 , x+2y> 12 and 3 x+2y> 24.

b Find all integer values ofxandyinRsuch that 2 x+3y=24.

c Find the minimum value of 2 x+5y for all (x,y) 2 Rwith integer coordinates.

d Find the minimum value of 12 x+8y for all (x,y) 2 Rwith integer coordinates.

State the coordinates of any point where this minimum value occurs.

IGCSE01

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Y:\HAESE\IGCSE01\IG01_32\645IGCSE01_32.CDR Friday, 31 October 2008 9:34:54 AM PETER