June 2022 Paper 1 Q13
13 Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor.
The entrance to the cave joins the floor at the points \(O\) and \(P\).
Garry models the shape of the cross section of the entrance to the cave using the equation
\[x^2 + y^2 = a\sqrt{x} - y\]where \(a\) is a constant, and \(x\) and \(y\) are the horizontal and vertical distances respectively, in metres, measured from \(O\).
(a) The distance \(OP\) is 16 metres.
Find the value of \(a\) that Garry should use in the model. [2 marks]
(b) Show that the maximum height of the cave above \(OP\) is approximately 10.5 metres. [6 marks]
(c) Suggest one limitation of the model Garry has used. [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(y\) = 0 and \(x\) = 16 correctly into \(x^2 + y^2 = a\sqrt{x} - y\) | M1 | 3.4 |
| Obtains \(a\) = 64 | A1 | 1.1b |
| (2) |
Typical solution
\[x^2 + y^2 = a\sqrt{x} - y\]\[16^2 + 0^2 = a\sqrt{16} - 0\]\[256 = 4a\]\[a = 64\]| Scheme | Marks | AO |
|---|---|---|
| Differentiates implicitly with either \(2y\dfrac{\mathrm{d}y}{\mathrm{d}x}\) or \(-\dfrac{\mathrm{d}y}{\mathrm{d}x}\) seen | B1 | 3.1b |
| Differentiates any two of the four terms correctly. Can be in terms of \(a\) or with their \(a\) value | M1 | 1.1a |
| Obtains a fully correct differentiated equation Can be in terms of \(a\) Follow through their \(a\) value \(2x + 2y\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{a}{2}x^{-\frac{1}{2}} - \dfrac{\mathrm{d}y}{\mathrm{d}x}\) | A1F | 1.1b |
| Uses \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) | M1 | 1.1a |
| Substitutes their numerical \(x\) value where \(0 \lt x \lt 16\), into the model with their \(a\) value | M1 | 3.4 |
| Obtains a value for \(y\) AWRT 10.51 and concludes that the maximum height is approximately 10.5 metres AG Condone equals Must state units CSO | R1 | 3.2a |
| (6) |
Typical solution
\[2x + 2y\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{64}{2}x^{-\frac{1}{2}} - \frac{\mathrm{d}y}{\mathrm{d}x}\]\[\frac{\mathrm{d}y}{\mathrm{d}x} = 0 \Rightarrow 2x = \frac{32}{\sqrt{x}}\]\[x^{\frac{3}{2}} = 16\]\[x = 6.3496\ldots\]\[(6.3496\ldots)^2 + y^2 = 64\sqrt{6.3496\ldots} - y\]\[y = 10.51\]Maximum height is approximately 10.5 metres
| Scheme | Marks | AO |
|---|---|---|
| States or infers that the entrance is unlikely to be a smooth curve Accept:
| E1 | 3.5b |
| (1) | ||
| (9 marks) |
Typical solution
The entrance to the cave is unlikely to be perfectly smooth