Problem

A company makes three products, P, Q and R.

Each product must go through two machines, Machine 1 and Machine 2.

The time taken on each machine, in minutes per unit, is shown in the table.

	P	Q	R
Machine 1	4	3	5
Machine 2	2	5	3

Machine 1 is available for 240 minutes per week.

Machine 2 is available for 210 minutes per week.

The profit from selling each unit of P is £6, each unit of Q is £7 and each unit of R is £10.

The company wishes to maximise the profit from the sale of these products.

Let $x$ be the number of units of P made per week, $y$ be the number of units of Q made per week and $z$ be the number of units of R made per week.

(a) State the objective function for this problem.

(b) Write down two inequalities, other than $x \geq 0$, $y \geq 0$, $z \geq 0$, that model the constraints of this problem.

(c) The company decides to make exactly 20 units of R per week. Show that the constraints reduce to

$2x + y \leq 70$ $x + 4y \leq 55$

(d) Draw the graph of these two inequalities, on the axes provided in the answer book, indicating clearly the feasible region.

(e) Hence determine the number of units of P and the number of units of Q that the company should make per week to maximise the profit.