D1 June 2008 Q8
8. Class 8B has decided to sell apples and bananas at morning break this week to raise money for charity. The profit on each apple is 20p, the profit on each banana is 15p. They have done some market research and formed the following constraints.
- They will sell at most 800 items of fruit during the week.
- They will sell at least twice as many apples as bananas.
- They will sell between 50 and 100 bananas.
Assuming they will sell all their fruit, formulate the above information as a linear programming problem, letting \(a\) represent the number of apples they sell and \(b\) represent the number of bananas they sell.
Write your constraints as inequalities. (7)
| Scheme | Marks |
|---|---|
| Maximise \((P=)\ 0.2a + 0.15b\) or \(20a + 15b\) o.e. | B1 B1 |
| (2) | |
| Subject to \[\begin{aligned} a + b &\leqslant 800\\ a &\geqslant 2b\\ 50 \leqslant b &\leqslant 100\\ a &\geqslant 0\end{aligned}\] | B1 B2, 1, 0 B1 B1 |
| (5) | |
| (7 marks) |
Notes
1B1: ‘Maximise’
2B1: ratio of coefficients correct
3B1: cao
4B1: ratio of coefficients of \(a\) and \(b\) correct.
5B1: inequality correct way round i.e. \(a \geqslant \ldots b\)
6B1: cao accept < – accept two separate inequalities here
7B1: cao
- Penalise < and > only once with last B mark earned
- Be generous on letters a, b, A, B, x, y etc and mixed, but remove last B mark earned if inconsistent or 3 letters in the ones marked.