D1 January 2013 Q6

EdexcelOld spec13 marksLinear Programming

6.

Figure 6: graph with the lines x = 60, y = 30 and a line labelled (a) through the origin, rejected regions shaded
Figure 6

Lethna is producing floral arrangements for an awards ceremony.
She will produce two types of arrangement, Celebration and Party.
Let \(x\) be the number of Celebration arrangements made.
Let \(y\) be the number of Party arrangements made.

Figure 6 shows three constraints, other than \(x, y \geqslant 0\)

The rejected region has been shaded.

Given that two of the three constraints are \(y \leqslant 30\) and \(x \leqslant 60\),

(a) write down, as an inequality, the third constraint shown in Figure 6. (2)

Each Celebration arrangement includes 2 white roses and 4 red roses.

Each Party arrangement includes 1 white rose and 5 red roses.

Lethna wishes to use at least 70 white roses and at least 200 red roses.

(b) Write down two further inequalities to represent this information. (3)
(c) Add two lines and shading to Diagram 1 in the answer book to represent these two inequalities. (2)
(d) Hence determine the feasible region and label it \(R\). (1)

The times taken to produce each Celebration arrangement and each Party arrangement are 10 minutes and 4 minutes respectively. Lethna wishes to minimise the total time taken to produce the arrangements.

(e) Write down the objective function, \(T\), in terms of \(x\) and \(y\). (1)
(f) Use point testing to find the optimal number of each type of arrangement Lethna should produce, and find the total time she will take. (4)
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