A2 June 2022 Q10
10.
Figure 2 shows a picture of a plant pot.
The plant pot has
- a flat circular base of radius 10 cm
- a height of 15 cm
Figure 3 shows a sketch of the curve \(C\) with parametric equations
\[x = 10 + 15t - 5t^3 \qquad y = 15t^2 \qquad 0 \leqslant t \leqslant 1\]The curved inner surface of the plant pot is modelled by the surface of revolution formed by rotating curve \(C\) through \(2\pi\) radians about the \(y\)-axis.
Each plant pot will be painted with one coat of paint, both inside and outside.
The paint in one tin will cover an area of 12 m2
| Scheme | Marks | AO |
|---|---|---|
| Surface of revolution \(= 2\pi\int x\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2 + \left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2}\,\mathrm{d}t\) \(= 2\pi\int (10 + 15t - 5t^3)\sqrt{(15 - 15t^2)^2 + (30t)^2}\,\mathrm{d}t\) | M1 A1 | 3.4 1.1b |
| Surface of revolution \(= \{2\pi\}\int \ldots \sqrt{225 - 450t^2 + 225t^4 + 900t^2}\,\mathrm{d}t\) \(= \{2\pi\}\int \ldots \sqrt{225 + 450t^2 + 225t^4}\,\mathrm{d}t\) | M1 | 1.1b |
| \(= \{2\pi\}\int \ldots \sqrt{(15 + 15t^2)^2}\,\mathrm{d}t\) \(= \{2\pi\}\int \ldots (15 + 15t^2)\,\mathrm{d}t\) | M1 | 2.1 |
| \(= 2\pi\int (10 + 15t - 5t^3)(15 + 15t^2)\,\mathrm{d}t\) \(\left\{2\pi\int (150 + 150t^2 + 225t + 225t^3 - 75t^3 - 75t^5)\,\mathrm{d}t\right\}\) \(150\pi\int_0^1 (2 + 3t + 2t^2 + 2t^3 - t^5)\,\mathrm{d}t\) * | A1* | 1.1b |
| (5) |
Notes
M1: Uses the formula surface area \(2\pi\int x\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2 + \left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2}\,\mathrm{d}t\) with their model.
A1: Correct formula for the inner surface area.
M1: Starts the process of manipulating into an integrable form by squaring and simplifying the expression under the square root.
M1: Completes the process by writing as something squared and cancelling.
A1*: Achieves the printed answer with no errors seen. Evidence of 75 taken out as a factor (it may be in stages) Limits just stated.
Note working with \(2\pi\int y\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2 + \left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2}\,\mathrm{d}t\) the maximum is M0A0M1M1A0
(corrected from the printed mark scheme: the surface area formula and the last line of (a) are printed in a garbled symbol font; they are typed here as intended)
| Scheme | Marks | AO |
|---|---|---|
| Surface of revolution \(= 150\pi\left[2t + \frac{3}{2}t^2 + \frac{2}{3}t^3 + \frac{2}{4}t^4 - \frac{1}{6}t^6\right]_0^1\) \(= 150\pi\left[\left(2(1) + \frac{3}{2}(1)^2 + \frac{2}{3}(1)^3 + \frac{2}{4}(1)^4 - \frac{1}{6}(1)^6\right) - (0)\right]\) | M1 | 1.1b |
| Surface of revolution \(= 675\pi =\) awrt 2120 | A1 | 1.1b |
| Adds their surface of revolution to their area of the circular base \(= \text{‘}675\pi\text{’} + \pi \times 10^2 = \ldots\) | M1 | 3.4 |
| Total surface area of the inner surface of the pot is \(775\pi\) or awrt 2430 | A1 | 1.1b |
| (4) |
Notes
M1: Integrates the polynomial and uses the limits of \(t = 0\) and \(t = 1\) the correct way around and subtracts. The limit of \(t = 0\) can be implied.
A1: Correct value for the inner surface area of revolution.
M1: Adds their area of the inner surface to their area of the circular base
A1: Correct total inner surface area for the plant pot.
Note Correct answer for the integration \(675\pi = 2120.6\) seen scores M1 A1
| Scheme | Marks | AO |
|---|---|---|
| Number of plant pots \(= \dfrac{120\,000}{2 \times \text{their total surface area}}\) or Number of plant pots \(= \dfrac{12}{2 \times \frac{\text{their total surface area}}{100^2}}\) | M1 | 3.4 |
| 24 (complete plant pots) | A1 | 1.1b |
| (2) |
Notes
M1: Finds the total number of plant pots by dividing 120 000 by 2 times their total surface area or divides 12 by 2 times their surface area/1002
A1: 24 (complete plant pots).
| Scheme | Marks | AO |
|---|---|---|
| For example: The surface area of the outside of the plant pot will be more than the inside. There will be a (small) rim on the plant pot which is not considered There is thickness to the pot The surface area of the pot may not perfectly fit the curve Pot may not be smooth | B1 | 3.5b |
| (1) | ||
| (12 marks) |
Notes
B1: Gives a correct limitation of the model.