D1 June 2016 Q8

EdexcelOld spec14 marksLinear Programming

8. Charlie needs to buy storage containers.

There are two different types of storage container available, standard and deluxe.

Standard containers cost £20 and deluxe containers cost £65. Let \(x\) be the number of standard containers and \(y\) be the number of deluxe containers.

The maximum budget available is £520

(a) Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. (1)

Three further constraints are:

\[\begin{aligned} x &\geqslant 2\\ -x + 24y &\geqslant 24\\ 7x + 8y &\leqslant 112 \end{aligned}\]
(b) Add lines and shading to Diagram 1 in the answer book to represent all four constraints. Hence determine the feasible region and label it R. (4)

The capacity of a deluxe container is 50% greater than the capacity of a standard container. Charlie wishes to maximise the total capacity.

(c) State an objective function, in terms of \(x\) and \(y\). (1)
(d) Use the objective line method to find the optimal vertex, V, of the feasible region. You must make your objective line clear and label the optimal vertex V. (3)
(e) Calculate the exact coordinates of vertex V. (2)
(f) Determine the number of each type of container that Charlie should buy. You must make your method clear and calculate the cost of purchasing the storage containers. (3)
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