7. Ian plans to produce two types of book, hardbacks and paperbacks. He will use linear programming to determine the number of each type of book he should produce. Let \(x\) represent the number of hardbacks Ian will produce. Let \(y\) represent the number of paperbacks Ian will produce. Each hardback takes 1 hour to print and 15 minutes to bind.
Each paperback takes 35 minutes to print and 24 minutes to bind.
The printing machine must be used for at least 14 hours. The binding machine must be used for at most 8 hours.

1. 1. Show that the printing time restriction leads to the constraint \(12 x + 7 y \geqslant k\), where \(k\) is a constant to be determined.
   2. Write the binding time restriction in a similar simplified form. Ian decides to produce at most twice as many hardbacks as paperbacks.
2. Write down an inequality to model this constraint in terms of \(x\) and \(y\).
3. Add lines and shading to Diagram 1 in the answer book to represent the constraints found in (a) and (b). Hence determine, and label, the feasible region R. Ian wishes to maximise \(\mathrm { P } = 60 x + 36 y\), where P is the total profit in pounds.
4. 1. Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must draw and clearly label your objective line and the vertex V .
   2. Determine the exact coordinates of V. You must show your working.
5. Given that P is Ian's expected total profit, in pounds, find the number of each type of book that he should produce and his maximum expected profit.