6. A company produces two types of self-assembly wooden bedroom suites, the 'Oxford' and the 'York'. After the pieces of wood have been cut and finished, all the materials have to be packaged. The table below shows the time, in hours, needed to complete each stage of the process and the profit made, in pounds, on each type of suite.
| \cline { 2 - 3 }
\multicolumn{1}{c|}{} | Oxford | York |
| Cutting | 4 | 6 |
| Finishing | 3.5 | 4 |
| Packaging | 2 | 4 |
| Profit \(( \pounds )\) | 300 | 500 |
The times available each week for cutting, finishing and packaging are 66, 56 and 40 hours respectively.
The company wishes to maximise its profit.
Let \(x\) be the number of Oxford, and \(y\) be the number of York suites made each week.
- Write down the objective function.
- In addition to
$$\begin{gathered}
2 x + 3 y \leq 33 ,
x \geq 0 ,
y \geq 0 ,
\end{gathered}$$
find two further inequalities to model the company’s situation. - On the grid in the answer booklet, illustrate all the inequalities, indicating clearly the feasible region.
- Explain how you would locate the optimal point.
- Determine the number of Oxford and York suites that should be made each week and the maximum profit gained.
(3)
It is noticed that when the optimal solution is adopted, the time needed for one of the three stages of the process is less than that available. - Identify this stage and state by how many hours the time may be reduced.
(3)