D1 January 2010 Q7
7. You are in charge of buying new cupboards for a school laboratory.
The cupboards are available in two different sizes, standard and large.
The maximum budget available is £1800. Standard cupboards cost £150 and large cupboards cost £300.
Let \(x\) be the number of standard cupboards and \(y\) be the number of large cupboards.
The cupboards will be fitted along a wall 9 m long. Standard cupboards are 90 cm long and large cupboards are 120 cm long.
Given also that \(y \geqslant 2\),
The capacity of a large cupboard is 40% greater than the capacity of a standard cupboard. You wish to maximise the total capacity.
| Scheme | Marks |
|---|---|
| \(x + 2y \leqslant 12 \quad (150x + 300y \leqslant 1800)\) | M1A1 |
| (2) |
Notes
1M1 – correct terms, accept = here, accept swapped coefficients.
1A1 – cao does not need to be simplified.
| Scheme | Marks |
|---|---|
| \(0.9x + 1.2y \leqslant 9\) | M1 |
| \(\rightarrow\ 3x + 4y \leqslant 30 \quad (*)\) | A1 cso |
| (2) |
Notes
1M1 – correct terms, must deal with cm/m correctly, accept = here.
1A1 – cso answer given.
| Scheme | Marks |
|---|---|
| (You need to buy) at least 2 large cupboards. | B1 |
| (1) |
Notes
1B1 – cao ‘at least’ and ‘2’ and ‘large’.
| Scheme | Marks |
|---|---|
| Capacity C and 140%C So total is \(Cx + \dfrac{140}{100}Cy\) | M1 |
| Simplify to \(7y + 5x \quad (*)\) | A1cso |
| (2) |
Notes
1M1 – ‘1.4’ or ‘5 x 40%’maybe ‘5+2’ seen, they must be seen to engage with 140% in some way.
1A1 – cso answer given.
| Scheme | Marks |
|---|---|
 | |
| Graph: \(y \geqslant 2\) | B1 |
| \(0.9x + 1.2y \leqslant 9 \quad (3x + 4y \leqslant 30)\) | B1 |
| \(x + 2y \leqslant 12 \quad (150x + 300y \leqslant 1800)\) | B1 |
| Lines labelled & drawn with a ruler | B1 |
| Shading, Region identified | B1, B1 |
| (6) |
Notes
Lines should be within 1 small square of correct point at axes.
1B1 – correctly drawing \(y = 2\).
2B1 – correctly drawing \(3x + 4y = 30\) [\(0.9x + 1.2y = 9\)]
3B1 – correctly drawing \(x + 2y = 12\) [\(150x + 300y = 1800\)], ft only if swapped coefficients in (a) (6,0) (2,8).
These next 3 marks are only available for candidates who have drawn at least 2 lines, including at least one ‘diagonal’ line with negative gradient.
4B1 – Ruler used. At least 2 lines labelled including one ‘diagonal’ line.
5B1 – Shading, or R correct, b.o.d. on their lines.
6B1 – all lines and R correct.
(Corrected from the printed mark scheme: the scheme prints \(0.9x + 1.2y \leqslant 12\) and \([0.9x + 1.2y = 12]\); the wall constraint from (b) is \(0.9x + 1.2y \leqslant 9\).)
| Scheme | Marks |
|---|---|
| Consider points and value of \(5x + 7y\): Or draw a clear profit line | M1A1 |
| (7,2) \(\rightarrow\) 49 or \((7\tfrac{1}{3}, 2) \rightarrow 50\tfrac{2}{3}\), or \((7.3, 2) \rightarrow 50.5\) (6,3) \(\rightarrow\) 51 (0,6) \(\rightarrow\) 42 | |
| (0,2) \(\rightarrow\) 14 | A1 |
| Best option is to buy 6 standard cupboards and 3 large cupboards. | A1 |
| (4) | |
| (17 marks) |
Notes
1M1 At least 2 points tested or objective line drawn with correct m or 1/m, minimum intercepts 3.5 and 2.5.
1A1 – 2 points correctly tested or objective line correct.
2A1 – 3 points correctly tested or objective line correct and distinct/labelled.
3A1 – 6 standard and 3 large, accept (6,3) if very clearly selected in some way.