October 2021 Paper 1 Q11
11 A balloon is being inflated. The balloon is modelled as a sphere with radius \(x\) cm at time \(t\) s. The volume \(V\,\text{cm}^3\) is given by \(V = \frac{4}{3}\pi x^3\).
The rate of increase of volume is inversely proportional to the radius of the balloon. Initially, when \(t = 0\), the radius of the balloon is 5 cm and the volume of the balloon is increasing at a rate of \(21\,\text{cm}^3\,\text{s}^{-1}\).
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}V}{\mathrm{d}t} = \dfrac{k}{x}\) | M1 | 3.3 |
| When \(t = 0\), \(x = 5\) and \(\dfrac{\mathrm{d}V}{\mathrm{d}t} = 21\), so \(\dfrac{\mathrm{d}V}{\mathrm{d}t} = \dfrac{105}{x}\) | A1 | 3.3 |
| \(\dfrac{\mathrm{d}V}{\mathrm{d}x} = 4\pi x^2\), | B1 | 2.1 |
| so the chain rule gives \(\dfrac{\mathrm{d}V}{\mathrm{d}t} = \dfrac{\mathrm{d}V}{\mathrm{d}x} \times \dfrac{\mathrm{d}x}{\mathrm{d}t} = 4\pi x^2 \dfrac{\mathrm{d}x}{\mathrm{d}t}\) | M1 | 2.1 |
| Hence \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{1}{4\pi x^2} \times \dfrac{105}{x} = \dfrac{105}{4\pi x^3}\) AG | A1 | 3.3 |
| [5] |
Notes
M1: Expresses inverse proportionality with a constant
A1: Evaluating \(k\), oe (may be done later)
B1: may be embedded in chain rule
M1: Use of the chain rule
A1: Convincing argument
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int 4\pi x^3\,\mathrm{d}x = \int 105\,\mathrm{d}t\) | M1 | 3.1a |
| \(\pi x^4 = 105t + c\) | A1 | 1.1b |
| When \(t = 0,\ x = 5\) so \(c = 625\pi\) | M1 A1 | 3.3 3.3 |
| When \(t = 120\) \(x = \sqrt[4]{\dfrac{105}{\pi} \times 120 + 625} = 8.25\) cm | A1 | 3.4 |
| [5] |
Notes
M1: Separating the variables
A1: Condone missing \(+c\) here
M1: Using initial conditions
A1: Correct value for \(c\)
A1: cao
| Scheme | Marks | AO |
|---|---|---|
| As \(t\) gets very large, the volume gets very large so the balloon will get beyond the maximum it can be without bursting and so burst. | E1 | 3.5b |
| [1] |
Notes
E1: Conveys the idea that \(t \to \infty \Rightarrow V \to \infty\) or \(x \to \infty\)
Indicates a practical problem with very large volume