4MA0 Higher January 2015 Paper 3R Q17
17
Diagram NOT accurately drawn
The diagram shows a cuboid of volume \(V\) cm3
The length of the cuboid is \(y\) cm
The width and height of the cuboid are both \(x\) cm
The total length of all the edges of the cuboid is 112 cm
(a) Show that \(V = 28x^2 - 2x^3\) (3)
(b) Find \(\dfrac{\mathrm{d}V}{\mathrm{d}x}\) (2)
(c) Find the maximum value of \(V\)
Give your answer correct to 3 significant figures. (3)
Give your answer correct to 3 significant figures. (3)
| Scheme | Marks |
|---|---|
| \(8x + 4y = 112\) | M1 |
| \(y = \dfrac{112 - 8x}{4}\) oe | M1 |
\(V = x^2 \times \dfrac{112 - 8x}{4}\) Answer: proof | A1 |
| (3) |
Notes
M1: correct equation linking \(x\) and \(y\)
M1: expression \(y\) in terms of \(x\)
A1: Conclusion with full working shown
| Scheme | Marks |
|---|---|
| \(56x - 6x^2\) | B2 |
| (2) |
Notes
B2: B1 for \(56x\); B1 for \(-6x^2\)
| Scheme | Marks |
|---|---|
| \(56x - 6x^2 = 0\) | M1 |
| \(x = \dfrac{28}{3}\) oe | A1 |
| 813 | A1 |
| (3) | |
| (8 marks) |
Notes
M1: ft for equating differential to 0
A1: Correct value for \(x\)
A1: for answer that rounds to 813