Linear Programming - part (b)

We looked at a few linear programming problems today.
There are 3 parts to tackling these questions.
STEP 1
Read through the text and extract 2 inequalities.
If you look back over the years, you will see that this question mostly comes up in a formulaic manner. In 2005 you were asked
"A manufacturer of garden furniture produces plastic chairs and tables. Each chair requires 2 kg of raw material and each table requires 5 kg of raw material. In any working period the raw material used cannot exceed 800 kg.
Each chair requires 4 minutes of machine time and each table requires 4 minutes of machine time. The total machine time available in any working period is 1000 minutes."
Identify the resctrictions: the manufacturer is restricted by amount of raw material they can use and by the minutes of machine time they have available.
x is going to be the number of chairs they manufacture, so as each chair reguires 2kg of raw material, they will require 2x kg for all the chairs they make (so if it turned out that they made 30 chairs, then they would need 2(30) = 60kg.)
Note that the words "cannot exceed" translate to "≤" in maths.
You build your two inequalities like this:
Raw materials: 2x + 5y ≤ 800
Time: 4x + 4y ≤ 1000

However in 2007, it was worded this way
"A developer is planning a holiday complex of cottages and apartments.
Each cottage will accommodate 3 adults and 5 children and each apartment will accommodate 2 adults and 2 children.
The other facilities in the complex are designed for a maximum of 60 adults and a maximum of 80 children."
Here the inequalities are not so obvious. A trick for working these out is to look at the two limiting factors - you are told that the max number of adults is 60 and the max number of children is 80.
So you know that your inequalities will be of the form
Adults: ...x + ....y ≤ 60
and
Children: ...x + ...y ≤ 80
(in other words the first inequality concerns restrictions of number of adults and the second inequality concerns restrictions of number of children)
Once you know that all the children info goes into the first inequality, you can find it more easily.

STEP 2
Graph the inequalities.
To do this you need to write the inequalities as equations. Then plot them the way you would in coordinate geometry - the easiest way is to use the intercept method (if you've forgotten this, it is described below in the section on Question 11 part(a).
Use the test-point to work out which half-plane you are interested in for each line. The resulting shaded in area is almost always a quadrilateral bounded by the x and y axes.

STEP 3
Evaluate an expression. After you've been asked to graph all the information you'll be told another piece of information about x and y, this time with no restriction.
In 2005 this was "The manufacturer sells each chair for €20 and each table for €40."
In 2007 it was "If the rental income per night will be €65 for a cottage and €40 for an apartment ...". In each case you are asked to maximise profit or revenue. You use this new information to create an expression in x and y.
In 2005 it would be 20x + 40y
In 2007 it would be 65x + 40y
The optimum values occur at the vertices (corners) of the shaded region. The vertex formed by the intersection of the two lines you just graphed must be calculated using simultaneous equations. The others can be read from the graph.
Then you take the (x,y) co-ordinates of each vertex and sub the x and y values into the expression.
Highlight the optimal value and very important, write the answer as an english sentence.
"The developer should build 10 cottages and 15 apartments".

There is usually a final part to this question - always something different, requiring you to evaluate another expression, find a percentage, deduct some amount from the overall profit etc.