D1 June 2013 Q6
6. Harry wants to rent out boats at his local park. He can use linear programming to determine the number of each type of boat he should buy.
Let \(x\) be the number of 2-seater boats and \(y\) be the number of 4-seater boats.
One of the constraints is
\[x + y \geqslant 90\]Another constraint is
\[2x \leqslant 3y\]A third constraint is
\[y \leqslant x + 30\]Each 2-seater boat costs £100 and each 4-seater boat costs £300 to buy. Harry wishes to minimise the total cost of buying the boats.
| Scheme | Marks |
|---|---|
| He must buy at least 90 boats in total (\(\boldsymbol{x + y \geqslant 90}\)) | B1 |
| (1) |
Notes
a1B1 CAO (must have ‘boats’, ‘least’, ‘90’, must be talking about boats not cost)
| Scheme | Marks |
|---|---|
| E.g. The number of 2-seater boats(\(x\)) must be less than or equal to 1.5 times the number of 4-seater boats (\(y\)). (check: \(y = 2,\ x = 3, 2, 1, \ldots\)) (\(\boldsymbol{2x \leqslant 3y}\)) E.g. The number of 4-seater boats (\(y\)) must be greater than or equal to 2/3 the number of 2-seater boats (\(x\)). (check: \(x = 3,\ y = 2, 3, 4, \ldots\)) | B1 B1 |
| (2) |
Notes
b1B1 For a statement in context with either the ratio of coefficients correct (the 2 with the 2-seater and the 3 with the 4-seater) or inequality correct with correct numbers present but not in the correct ratio.
b2B1 Clear accurate correct statement in context.
Examples for part (b) scoring B1 B1 (useful check: when \(y = 2,\ x = 3, 2, 1, \ldots\) or when \(x = 3,\ y = 2, 3, 4, \ldots\))
- Twice the number of 2-seater boats must be at most three times the number of 4-seater boats
- Three times the number of 4-seater must be at least twice the number of 2-seater boats
- For every three 2-seater boats there must be at least two 4-seater boats (or multiple of this ratio)
- For every two 4-seater boats there must be at most three 2-seater boats (or multiple of this ratio)
- At most 60% of the total boats are 2-seater
- At least 40% of the total boats are 4-seater
Examples of B1 B0 – in each case either the inequality is the correct way round OR the 2 is with 2-seater boats and the 3 is with the 4-seater boats (accept multiples of 2 and 3) (useful numbers: when \(y = 2,\ x = 3, 4, 5, \ldots\) when \(x = 3,\ y = 2, 1, \ldots\), when \(y = 3,\ x = 2, 1, \ldots\), when \(x = 2,\ y = 3, 4, 5, \ldots\))
- Twice the number of 2-seater boats must be at least three times the number of 4-seater boats
- Three times the number of 4-seater must be at most twice the number of 2-seater boats
- Three times the number of 2-seater must be at least twice the number of 4-seater boats
- For every three 2-seater boats there must be at most two 4-seater boats (or multiple of this ratio)
- For every two 4-seater boats there must be at least three 2-seater boats (or multiple of this ratio)
- For every two 2-seater boats there must be at least three 4-seater boats (or multiple of this ratio)
- For every three 4-seater boats there must be at most two 2-seater boats (or multiple of this ratio)
- At least 60% of the total boats are 2-seater
- At most 40% of the total boats are 4-seater
- At least 60% of the total boats are 4-seater
- At most 40% of the total boats are 2-seater
| Scheme | Marks |
|---|---|
 | |
| The correct 3 lines added; \(x + y = 90\); \(3y = 2x\); \(y = x + 30\) | B1; B1; B1 |
| Region, R labelled | B1 |
| (4) |
Notes
c1B1 \(x + y = 90\) correctly drawn. Must pass within one small square of the points of intersection with the axes
c2B1 \(3y = 2x\) correctly drawn. Must pass within one small square of the origin and (90, 60).
c3B1 \(y = x + 30\) correctly drawn. Must pass within one small square of (0, 30) and (60, 90).
c4B1 Region, R, correctly labelled – not just implied by shading – must have scored all three previous marks in this part.
| Scheme | Marks |
|---|---|
| (minimise C = ) \(100x + 300y\) | B1 |
| (1) |
Notes
d1B1 CAO (isw if \(100x + 300y\) ‘simplified’ to \(k(100x + 300y)\) but if \(100x + 300y\) not stated then B0)
| Scheme | Marks |
|---|---|
| Method clear – either at least 2 vertices tested or objective line drawn | M1 A1 |
| (54, 36), so 54 2-seater and 36 4-seater | B1 |
| At a cost of £16 200 | B1 |
| (4) | |
| (12 marks) |
Notes
e1M1 Line must be correct to within one small square if extended from axis to axis OR attempting to find two vertices of their R (or the correct R) by either reading off their graph or using simultaneous equations and testing using their objective function.
e1A1 Correct objective line (same condition that the line must be correct to within one small square if extended from axis to axis) OR testing (30, 60) correctly (giving 21 000) and testing (54, 36) correctly (giving 16 200).
e1B1 Correct point identified. (Condone in terms of \(x\) and \(y\) rather than in terms of boats.)
e2B1 CAO – condone lack of/incorrect units on the cost.