June 2018 Paper 2 Q10
10. A spherical mint of radius 5 mm is placed in the mouth and sucked.
Four minutes later, the radius of the mint is 3 mm.
In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius.
Using this model and all the information given,
(You should define the variables that you use.) (5)
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}r}{\mathrm{d}t} \propto \pm\dfrac{1}{r^2}\) or \(\dfrac{\mathrm{d}r}{\mathrm{d}t} = \pm\dfrac{k}{r^2}\) (for \(k\) or a numerical \(k\)) | M1 | 3.3 |
| \(\displaystyle\int r^2\,\mathrm{d}r = \int \pm k\,\mathrm{d}t \Rightarrow \ldots\) (for \(k\) or a numerical \(k\)) | M1 | 2.1 |
| \(\dfrac{1}{3}r^3 = \pm kt\ \{+ c\}\) | A1 | 1.1b |
| \(t = 0, r = 5\) and \(t = 4, r = 3\) gives \(\dfrac{1}{3}r^3 = -\dfrac{49}{6}t + \dfrac{125}{3}\), where \(r\), in mm, is the radius {of the mint} and \(t\), in minutes, is the time from when it {the mint} was placed in the mouth or \(t = 0, r = 5\) and \(t = 240, r = 3\) gives \(\dfrac{1}{3}r^3 = -\dfrac{49}{360}t + \dfrac{125}{3}\), where \(r\), in mm, is the radius {of the mint} and \(t\), in seconds, is the time from when it {the mint} was placed in the mouth | M1 A1 | 3.1a 1.1b |
| (5) |
Notes
M1: Translates the description of the model into mathematics. See scheme.
M1: Separates the variables of their differential equation which is in the form \(\dfrac{\mathrm{d}r}{\mathrm{d}t} = \mathrm{f}(r)\) and some attempt at integration. (e.g. attempts to integrate at least one side).
e.g. \(\displaystyle\int r^2\,\mathrm{d}r = \int \pm k\,\mathrm{d}t\) and some attempt at integration.
Condone the lack of integral signs
Note: You can imply the M1 mark for \(r^2 dr = -k\,\mathrm{d}t \Rightarrow \dfrac{1}{3}r^3 = -kt\)
Note: A numerical value of \(k\) (e.g. \(k = \pm 1\)) is allowed for the first two M marks
A1: Correct integration to give \(\dfrac{1}{3}r^3 = \pm kt\) with or without a constant of integration, \(c\)
M1: For a complete process of using the boundary conditions to find both their unknown constants and finds an equation linking \(r\) and \(t\)
So applies either
- \(t = 0, r = 5\) and \(t = 4, r = 3\), or
- \(t = 0, r = 5\) and \(t = 240, r = 3\),
on their integrated equation to find their constants \(k\) and \(c\) and obtains an equation linking \(r\) and \(t\)
A1: Correct equation, with variables \(r\) and \(t\) fully defined including correct reference to units.
- \(\dfrac{1}{3}r^3 = -\dfrac{49}{6}t + \dfrac{125}{3}\), {or an equivalent equation,} where \(r\), in mm, is the radius {of the mint} and \(t\), in minutes, is the time from when it {the mint} was placed in the mouth
- \(\dfrac{1}{3}r^3 = -\dfrac{49}{360}t + \dfrac{125}{3}\), {or an equivalent equation,} where \(r\), in mm, is the radius {of the mint} and \(t\), in seconds, is the time from when it {the mint} was placed in the mouth
Note: Allow correct equations such as
- in minutes, \(r = \sqrt[3]{\dfrac{250 - 49t}{2}}\), \(r^3 = -\dfrac{49}{2}t + 125\) or \(t = \dfrac{250 - 2r^3}{49}\)
- in seconds, \(r = \sqrt[3]{\dfrac{15000 - 49t}{120}}\), \(r^3 = -\dfrac{49}{120}t + 125\) or \(t = \dfrac{15000 - 120r^3}{49}\)
Note: \(t\) defined as “the time from the start” is not sufficient for the final A1
| Scheme | Marks | AO |
|---|---|---|
| \(r = 0 \Rightarrow 0 = -\dfrac{49}{6}t + \dfrac{125}{3} \Rightarrow 0 = -49t + 250 \Rightarrow t = \ldots\) | M1 | 3.4 |
| time = 5 minutes 6 seconds | A1 | 1.1b |
| (2) |
Notes
M1: Sets \(r = 0\) in their part (a) equation which links \(r\) with \(t\) and rearranges to make \(t = \ldots\)
A1: 5 minutes 6 seconds cao (Note: 306 seconds with no reference to 5 minutes 6 seconds is A0)
Note: Give M0 if their equation would solve to give a negative time or a negative time is found
Note: You can mark part (a) and part (b) together
| Scheme | Marks | AO |
|---|---|---|
Suggests a suitable limitation of the model. E.g.
| B1 | 3.5b |
| (1) | ||
| (8 marks) |
Notes
B1: See scheme
Note: Do not accept by itself
- mint may not dissolve at a constant rate
- rate of decrease of mint must be constant
- \(0 \leqslant t \lt \dfrac{250}{49},\ r \geqslant 0;\) without any written explanation
- reference to a mint having \(r \gt 5\)