D1 June 2017 Q7
7. A caterer can make three different sizes of salad; small, medium and large.
The caterer will make a total of at least 280 salads.
The caterer wants at least 35% of the salads to be small and no more than 20% of the salads to be large.
The caterer has enough ingredients to make 400 small salads or 300 medium salads or 200 large salads.
The profit on each small, medium and large salad is 40p, 60p and 85p respectively. The caterer wants to maximise his total profit.
Let \(x\) represent the number of small salads, \(y\) represent the number of medium salads and \(z\) represent the number of large salads.
Formulate this information 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. (8)
| Scheme | Marks |
|---|---|
| Maximise \((P =)\ 40x + 60y + 85z\) | B1 |
| Subject to: \(x + y + z \geqslant 280\) | B1 |
| \(\dfrac{7}{20}(x + y + z) \leqslant x\) which simplifies to \(13x \geqslant 7y + 7z\) | M1 A1 |
| \(\dfrac{1}{5}(x + y + z) \geqslant z\) which simplifies to \(x + y \geqslant 4z\) | M1 A1 |
| \(\dfrac{x}{400} + \dfrac{y}{300} + \dfrac{z}{200} \leqslant 1\) which simplifies to \(3x + 4y + 6z \leqslant 1200\) | M1 A1 |
| \((x,\ y,\ z \geqslant 0)\) | |
| (8 marks) |
Notes
1B1: Expression correct (or \(0.4x + 0.6y + 0.85z\)) together with ‘maximise’ or ‘max’ but not ‘maximum’ – isw if coefficients are subsequently simpified but either \(40x + 60y + 85z\) or \(0.4x + 0.6y + 0.85z\) must be seen at some point for this mark to be awarded
2B1: CAO
1M1: Correct method: \(\dfrac{7}{20}(x + y + z) \bullet x\) where \(\bullet\) is any inequality or =. The bracket must be present or implied by later working. An exact equivalent answer (with or without integer coefficients but with correct inequality sign) with no working can score M1. Accept equivalent fractions or decimals for 7/20 but not 35% (unless later converted to a correct fraction/decimal)
1A1: CAO – answer must have integer coefficients with like terms collected i.e. \(k(13x \geqslant 7y + 7z)\) for any positive integer \(k\) - the correct answer with no working can score M1 A1
2M1: Correct method: \(\dfrac{1}{5}(x + y + z) \bullet z\) where \(\bullet\) is any inequality or = . The bracket must be present or implied by later working. An exact equivalent answer (with or without integer coefficients but with correct inequality sign) with no working can score M1. Accept equivalent fractions or decimals for 1/5 but not 20% (unless later converted to a correct fraction/decimal)
2A1: CAO – answer must have integer coefficients with like terms collected i.e. \(k(x + y \geqslant 4z)\) for any positive integer \(k\) - the correct answer with no working can score M1 A1
3M1: Correct complete method: \(\dfrac{x}{400} + \dfrac{y}{300} + \dfrac{z}{200} \bullet 1\) (oe) where \(\bullet\) is any inequality or =. An exact equivalent answer (with or without integer coefficients but with correct inequality sign) with no working can score M1
3A1: CAO – answer must have integer coefficients with like terms collected i.e. \(k(3x + 4y + 6z \leqslant 1200)\) for any positive integer \(k\) - the correct answer with no working can score M1 A1
Condone \(s\), \(m\) and \(l\) for \(x\), \(y\) and \(z\) for full marks – any other letter used then please send to review (unless clearly defined and then award as per the scheme)