A2 June 2022 Q6
6.
The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \dfrac{1}{3}x\). This shaded region is rotated through \(360^\circ\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm.
The cone is non-uniform and the mass per unit volume of the cone at the point \((x, y, z)\) is \(\lambda x\ \text{kg cm}^{-3}\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.
A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm, so that the centres of the two plane surfaces coincide.
The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(kW\) newtons.
When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
| Scheme | Marks | AO |
|---|---|---|
| Mass of cone \(= \displaystyle\int_0^9 \pi y^2\lambda x\,\mathrm{d}x = \pi\lambda\int_0^9 \dfrac{x^3}{9}\,\mathrm{d}x\) | M1 | 3.4 |
| \(= \pi\lambda\left[\dfrac{x^4}{36}\right]_0^9 \qquad \left(= \dfrac{729\pi\lambda}{4}\,(\text{kg})\right)\) | A1 | 1.1b |
| Moments: \(\displaystyle\int_0^9 \pi y^2\lambda x\times x\,\mathrm{d}x = \pi\lambda\int_0^9 \dfrac{x^4}{9}\,\mathrm{d}x\) | M1 | 3.4 |
| \(= \dfrac{\pi\lambda}{45}\left[x^5\right]_0^9 \qquad \left(= \dfrac{\pi\lambda}{5}\times 9^4\right)\) | A1 | 1.1b |
| \(\Rightarrow d = \dfrac{\dfrac{\pi\lambda}{5}\times 9^4}{\dfrac{\pi\lambda}{4}\times 9^3}\) | DM1 | 2.1 |
| \(d = \dfrac{36}{5} = 7.2\,(\text{cm})\) | A1 | 1.1b |
| (6) |
Notes
NB: Some candidates are confusing the mass and the volume. For the first M1A1:
- If they have a correct method for the mass and they tell you that this is mass, award the marks.
- If they have a correct method for the mass say nothing, but use it correctly, award the marks.
- If they have a correct method for the mass, say nothing, and use it as the moment, then M0 because this implies that they do not think it is the mass.
M1: Use the model to find the mass of the cone. Allow without limits.
A1: Correct integration. Correct limits seen or implied
Substitution not required.
Allow 2/2 if π not seen and consistent with (b) if attempted
M1: Use the model to find the moment of the cone (usual rules for integration)
Allow without limits
A1: Correct integration. Correct limits seen or implied
Substitution not required.
Allow 2/2 if π not seen and consistent with (a)
M1: Complete method to find the distance of the centre of mass from the vertex. A complete method requires the two preceding M marks.
They need to get as far as a value for \(d\).
If they have a method that comes directly to this stage you might not see the \(\lambda\) or \(\pi\)
A1: Correct only
If all you see is \(\Rightarrow d = \dfrac{9^5}{45}\div\dfrac{9^4}{36}\) or even \(\Rightarrow d = \dfrac{9}{5}\times 4\) then award 6/6
Allow 6/6 if π not seen throughout but otherwise correct
| Scheme | Marks | AO |
|---|---|---|
| Remains at rest \(\Rightarrow\) centre of mass at centre of plane surface | B1 | 2.1 |
| Moments about diameter of plane surface: | M1 | 3.1b |
| \((9 - d)W\left\{= \left(9 - \dfrac{36}{5}\right)W\right\} = \dfrac{3}{8}\times 3\times kW\) | A1ft | 1.1b |
| \(k = \dfrac{8}{5}\) | A1 | 1.1b |
| (4) | ||
| (10 marks) |
Notes
B1: Correct deduction for location of c of m
Stated or implied by their moments equation
M1: Moments about diameter of plane face(s)
M0 if the moments equation contradicts the centre of mass being on the interface
M0 if using volume in place of mass
A1ft: Correct unsimplified equation. Follow their 7.2
Alternative moments equations:
Using vertex: \(W\bar{x} + kW\left(9 + \dfrac{3}{8}\times 3\right) = (W + kW)\times 9\)
Using base: \(W(12 - \bar{x}) + kW\left(3 - 3\times\dfrac{3}{8}\right) = (W + kW)3\)
If they are working with the axis at an angle they will possibly have trig terms which should cancel.
A1: Correct only