6 [Figure 1, printed on the insert, is provided for use in this question.]
A factory makes two types of lock, standard and large, on a particular day.
On that day:
the maximum number of standard locks that the factory can make is 100 ;
the maximum number of large locks that the factory can make is 80 ;
the factory must make at least 60 locks in total;
the factory must make more large locks than standard locks.
Each standard lock requires 2 screws and each large lock requires 8 screws, and on that day the factory must use at least 320 screws.
On that day, the factory makes \(x\) standard locks and \(y\) large locks.
Each standard lock costs \(\pounds 1.50\) to make and each large lock costs \(\pounds 3\) to make.
The manager of the factory wishes to minimise the cost of making the locks.
- Formulate the manager's situation as a linear programming problem.
- On Figure 1, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.
- Find the values of \(x\) and \(y\) that correspond to the minimum cost. Hence find this minimum cost.