D1 June 2014 (R) Q8
8. A manufacturer of frozen yoghurt is going to exhibit at a trade fair. He will take two types of frozen yoghurt, Banana Blast and Strawberry Scream.
He will take a total of at least 1000 litres of yoghurt.
He wants at least 25% of the yoghurt to be Banana Blast. He also wants there to be at most half as much Banana Blast as Strawberry Scream.
Each litre of Banana Blast costs £3 to produce and each litre of Strawberry Scream costs £2 to produce. The manufacturer wants to minimise his costs.
Let \(x\) represent the number of litres of Banana Blast and \(y\) represent the number of litres of Strawberry Scream.
Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients.
You should not attempt to solve the problem. (6)
| Scheme | Marks |
|---|---|
| Minimise \(C = 3x + 2y\) | B1 |
| Subject to: | |
| \(x + y \geqslant 1000\) | B1 |
| \(\dfrac{1}{4}(x + y) \leqslant x,\) simplifies to \(y \leqslant 3x\) | M1 A1 |
| \(2x \leqslant y\) | M1 A1 |
| \((x,\ y \geqslant 0)\) | |
| (6 marks) |
Notes
1B1: CAO – expression correct and ‘minimise’.
2B1: CAO
1M1: Correct method – must see \(\frac{1}{4}(x + y) \blacksquare x\) where \(\blacksquare\) is any inequality or =. The bracket must be present or implied by later working.
1A1: CAO – simplified – answer must have integer coefficients.
2M1: Correct method – one of \(2x \blacksquare y\) or \(x \blacksquare 2y\) where \(\blacksquare\) is any inequality or =.
2A1: CAO – answer must have integer coefficient.