AS June 2018 Paper 1 Q9
9
(a) Sketch the graph of \(y^2 = 4x\) [1 mark]
(b) Ben is using a 3D printer to make a plastic bowl which holds exactly \(1000\,\text{cm}^3\) of water.
Ben models the bowl as a region which is rotated through \(2\pi\) radians about the \(x\)-axis.
He uses the finite region enclosed by the lines \(x = d\) and \(y = 0\) and the curve with equation \(y^2 = 4x\) for \(y \geqslant 0\)
Ben models the bowl as a region which is rotated through \(2\pi\) radians about the \(x\)-axis.
He uses the finite region enclosed by the lines \(x = d\) and \(y = 0\) and the curve with equation \(y^2 = 4x\) for \(y \geqslant 0\)
(i) Find the depth of the bowl to the nearest millimetre. [4 marks]
(ii) What assumption has Ben made about the bowl? [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| Sketches correct parabola | B1 | 1.2 |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains \(\pi\displaystyle\int 4x\,\mathrm{d}x\) Limits not required for this mark. Condone missing \(\mathrm{d}x\) | M1 | 3.3 |
| Obtains \(2x^2\) and uses limits of \(d\) and \(0\) (oe). | B1 | 1.1b |
| Forms an equation of the form \(kd^2 = \text{volume}\) (oe) | M1 | 3.4 |
| Correct depth to nearest millimetre. Condone 126 or 12.6 without units. NMS: 126 or 12.6 scores 4/4. Using \(1000000\,\text{mm}^3\) leads to a correct answer of 399 mm for 4/4. | A1 | 3.2a |
| (ii) States appropriate assumption | B1 | 3.5b |
| (6 marks) |
Typical solution
(i)
\[\text{volume} = \pi\int_0^d y^2\,\mathrm{d}x\]\[\therefore 1000 = \pi\int_0^d 4x\,\mathrm{d}x\]\[\frac{1000}{\pi} = \left[\frac{4x^2}{2}\right]_0^d\]\[\frac{1000}{\pi} = 2d^2 - 0\]\[2\pi d^2 = 1000\]\[d = \sqrt{\frac{500}{\pi}}\]depth = 12.6 cm
(ii)
The thickness of the plastic is negligible