8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares \(x\) Luxury hampers and \(y\) Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:

• she will need to prepare at least 5 hampers of each size
• she will prepare at most a total of 32 hampers
• she will prepare at least twice as many Luxury hampers as Special hampers.

Each Luxury hamper that Nerys prepares makes her a profit of \(\pounds 15\); each Special hamper makes a profit of \(\pounds 20\). Nerys wishes to maximise her daily profit, \(\pounds P\).

1. Show that \(x\) and \(y\) must satisfy the inequality \(2 x + 3 y \leqslant 72\).
2. In addition to \(x \geqslant 5\) and \(y \geqslant 5\), write down two more inequalities that model the constraints above.
3. On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
4. 1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
   2. Calculate this profit.
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