Graphical Solution of LP problems (Edexcel A Level Further Maths: Decision Maths 1)

A shop sells two types of watch, analogue watches and digital watches.

The shop manager knows that, each month, she should order at least 60 watches in total.
In addition, at most 80% of the watches she orders must be digital.

Let be the number of analogue watches ordered and let be the number of digital watches ordered.

Write down inequalities, in terms of , to model these constraints.

Two further constraints are

Represent all these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, R.

The cost to the shop of ordering an analogue watch is five times the cost of ordering a digital watch. The shop manager wishes to minimise the total cost.

Determine the number of each type of watch the shop manager should order.
You must make your method clear.

Given that the minimum total cost of ordering the watches is £4455

Determine the cost of ordering one analogue watch and the cost of ordering one digital watch. You must make your method clear.

Figure 3

Figure 3 shows the constraints of a linear programming problem in and , where is the feasible region. Figure 3 also shows an objective line for the problem and the optimal vertex, which is labelled as .

The value of the objective at is 556

Express the linear programming problem in algebraic form. List the constraints as simplified inequalities with integer coefficients and determine the objective.

Figure 3

Figure 3 shows the constraints of a linear programming problem in  and , where  is the feasible region.

Write down the inequalities that define .

The objective is to maximise , where 

Obtain the exact value of  at each of the three vertices of  and hence find the optimal vertex, .

The objective is changed to maximise $Q$, where $Q=3x+ay$. Given that $a$ is a constant and the optimal vertex is still $V$, find the range of possible values of $a$.