An engineering firm makes two types of tractor, the ‘Rome’ and the ‘Brussels’. The number of each made per week is limited by storage capacity on site, the number of workers employed and manufacturing costs, as detailed below:

The maximum total storage capacity for a week’s production is for \(10\) tractors. However, there is not enough room to store more than \(8\) Brussels tractors from any week’s production and, in order to meet demand, at least \(3\) Rome tractors must be made each week.

In any given week, \(5\) workers are required to make one Rome tractor and \(2\) are needed to make one Brussels tractor. There are \(40\) workers altogether, but not all of them have to be used. It costs \(£16\,000\) to make a Rome tractor and \(£12\,000\) to make a Brussels tractor. A total of \(£144\,000\) is available each week.

  1. Using the symbols \(R\) for the number of Rome and \(B\) for the number of Brussels tractors made per week, write down \(5\) inequalities arising from the information above.

  2. Illustrate these inequalities on a graph in which the horizontal axis represents the number of Rome tractors made per week and the vertical axis represents the number of Brussels tractors made per week. Show the feasible region clearly.

  3. The profit on one Rome tractor is \(£3\,000\) and the profit on one Brussels tractor is \(£2\,000\). Find the number of each type which should be made weekly in order to maximise the profit, and find the maximum profit.