4MA0 Higher January 2012 Paper 3 Q14
14
The diagram shows a rectangular photo frame of area \(A\) cm\(^2\).
The width of the photo frame is \(x\) cm.
The height of the photo frame is \(y\) cm.
The perimeter of the photo frame is 72 cm.
(a) Show that \(A = 36x - x^2\) (3)
(b) Find \(\dfrac{\mathrm{d}A}{\mathrm{d}x}\) (2)
(c) Find the maximum value of \(A\). (3)
| Scheme | Marks |
|---|---|
| \(y = 36 - x\) | M2 |
| (Area =) \(x\,(36 - x)\) | A1 |
| (3) |
Notes
M2: M1 for \(x + y = 36\) oe or \(2y = 72 - 2x\)
A1: Must see \(x\) times \((36 - x)\) dep on M2
| Scheme | Marks |
|---|---|
| \((\mathrm{d}A/\mathrm{d}x) = 36 - 2x\) | B1 B1 |
| (2) |
Notes
B1 B1: B1 for 36 B1 for \(-2x\)
| Scheme | Marks |
|---|---|
| “\(36 - 2x\)” = 0 | M1 |
| \(x = 18\) | A1ft |
| (Area =) 324 | A1ft |
| (3) | |
| (Total 8 marks) |
Notes
M1: allow ft only on \(a + bx\) (\(a, b \neq 0\))