8. A shop sells three types of pen. These are ballpoint pens, rollerball pens and fountain pens. The shop manager knows that each week she should order

• at least 50 pens in total
• at least twice as many rollerball pens as fountain pens

In addition,

• at most \(60 \%\) of the pens she orders must be ballpoint pens
• at least a third of the pens she orders must be rollerball pens

Each ballpoint pen costs \(\pounds 2\), each rollerball pen costs \(\pounds 3\) and each fountain pen costs \(\pounds 5\)
The shop manager wants to minimise her costs.
Let \(x\) represent the number of ballpoint pens ordered, let \(y\) represent the number of rollerball pens ordered and let \(z\) represent the number of fountain pens ordered.

1. Formulate this information as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients. The shop manager decides to order exactly 10 fountain pens. This reduces the problem to the following $$\begin{array} { l r } \text { Minimise } & P = 2 x + 3 y
   \text { subject to } & x + y \geqslant 40
   & 2 x - 3 y \leqslant 30
   - x + 2 y \geqslant 10
   & y \geqslant 20
   & x \geqslant 0 \end{array}$$
2. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R .
3. 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.
4. Write down the number of each type of pen that the shop manager should order. Calculate the cost of this order.
   (Total \(\mathbf { 1 6 }\) marks)