C4 June 2012 Q2
2.
Figure 1 shows a metal cube which is expanding uniformly as it is heated.
At time \(t\) seconds, the length of each edge of the cube is \(x\) cm, and the volume of the cube is \(V\,\text{cm}^3\).
(a) Show that \(\dfrac{\mathrm{d}V}{\mathrm{d}x} = 3x^2\) (1)
Given that the volume, \(V\,\text{cm}^3\), increases at a constant rate of \(0.048\,\text{cm}^3\,\text{s}^{-1}\),
(b) find \(\dfrac{\mathrm{d}x}{\mathrm{d}t}\), when \(x = 8\) (2)
(c) find the rate of increase of the total surface area of the cube, in \(\text{cm}^2\,\text{s}^{-1}\), when \(x = 8\) (3)
| Scheme | Marks |
|---|---|
| \(V = x^3 \Rightarrow \dfrac{\mathrm{d}V}{\mathrm{d}x} = 3x^2\ \ *\) cso | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{\mathrm{d}x}{\mathrm{d}V} \times \dfrac{\mathrm{d}V}{\mathrm{d}t} = \dfrac{0.048}{3x^2}\) | M1 |
| At \(x = 8\) \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{0.048}{3\left(8^2\right)} = 0.00025 \quad \left(\text{cm}\,\text{s}^{-1}\right)\) \(2.5 \times 10^{-4}\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(S = 6x^2 \Rightarrow \dfrac{\mathrm{d}S}{\mathrm{d}x} = 12x\) | B1 |
| \(\dfrac{\mathrm{d}S}{\mathrm{d}t} = \dfrac{\mathrm{d}S}{\mathrm{d}x} \times \dfrac{\mathrm{d}x}{\mathrm{d}t} = 12x\left(\dfrac{0.048}{3x^2}\right)\) | M1 |
| At \(x = 8\) \(\dfrac{\mathrm{d}S}{\mathrm{d}t} = 0.024 \quad \left(\text{cm}^2\,\text{s}^{-1}\right)\) | A1 |
| (3) | |
| (6 marks) |