AS June 2025 Paper 1 Q7
7.
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Figure 1 shows a sketch of a hot air balloon.
When filled with air the balloon has a height of 24 metres.
Figure 2 shows a sketch of the curve \(C\) with equation
\[350x^2 = (12 + y)^2\left(A - y^2\right) \qquad x \geqslant 0\]where \(A\) is a constant.
The balloon is modelled by rotating \(C\) through 360° about the \(y\)-axis.
Given that one \(y\) intercept of \(C\) is \(-12\)
| Scheme | Marks | AO |
|---|---|---|
| \(x = 0\) and \(y = 12\) \(0 = (12 + 12)^2\left(A - 12^2\right) \Rightarrow A = 144\) | B1 | 3.3 |
| (1) |
Notes
B1: Uses \(x = 0\) and \(y = 12\) in the full equation to show that \(A = 144\)
Note Uses \(x = 0\) and \(y = -12\) is B0
| Scheme | Marks | AO |
|---|---|---|
| \(V = \dfrac{\pi}{350}\displaystyle\int_{-12}^{12} (12 + y)^2\left(144 - y^2\right)\ \{\mathrm{d}y\}\) | B1 | 3.4 |
| \((12 + y)^2\left(144 - y^2\right) = \ldots\left\{20736 + 3456y - 24y^3 - y^4\right\}\) \(V = \left\{\dfrac{\pi}{350}\right\}\displaystyle\int \left(20736 + 3456y - 24y^3 - y^4\right)\mathrm{d}y\) \(= \left\{\dfrac{\pi}{350}\right\}\left(20736y + 1728y^2 - 6y^4 - \dfrac{1}{5}y^5\right)\) \(= \{\pi\}\left(\dfrac{10368}{175}y + \dfrac{864}{175}y^2 - \dfrac{3}{175}y^4 - \dfrac{1}{1750}y^5\right)\) Alternative by parts \((12 + y)^2\left(144 - y^2\right) = (12 + y)^3(12 - y)\) \(V = \displaystyle\int (12 + y)^3(12 - y)\,\mathrm{d}y\) \(= \dfrac{1}{4}(12 + y)^4(12 - y) + \displaystyle\int \frac{1}{4}(12 + y)^4\,\mathrm{d}y\) \(= \dfrac{1}{4}(12 + y)^4(12 - y) + \dfrac{1}{20}(12 + y)^5\) | M1 A1 | 1.1b 1.1b |
| \(\left[20736(12) + 1728(12)^2 - 6(12)^4 - \dfrac{1}{5}(12)^5\right] -\) \(\left[20736(-12) + 1728(-12)^2 - 6(-12)^4 - \dfrac{1}{5}(-12)^5\right]\) \(= (323481.6) - (-74649.6)\) Alternative by parts \(\left[\dfrac{1}{4}(12 + 12)^4(12 - 12) + \dfrac{1}{20}(12 + 12)^5\right] - \left[\dfrac{1}{4}(12 - 12)^4(12 + 12) + \dfrac{1}{20}(12 - 12)^5\right]\) | M1 | 3.4 |
| \(= 3600\text{ m}^3\) cao | A1 | 2.2b |
| (5) |
Notes
(Corrected from the printed mark scheme: the value at the lower limit is printed as \(-74659.6\), here and in the notes; the correct value is \(-74649.6\).)
B1: Uses the model to set up the volume for the balloon, with limits, the d\(y\) may be implied
M1: Multiplies out the brackets and integrates \(\int x^n\,\mathrm{d}x \rightarrow x^{n+1}\). Alternatively uses integration by parts the correct way. Condone a slip when multiplying out
A1: Correct integration
M1: Uses the limits of \(-12\) and 12, subtracts the correct way round.
If the integration is correct this can be evidenced by for example \((323481.6) - (-74649.6)\) or 2903.56 − (− 670.05) if including \(\dfrac{\pi}{350}\)
If the integration is incorrect we must see the substitution of 12 and \(-12\) into their integrated function
Candidates may use limits of \(-12\) to 0 and then 0 to 12 and add which is fine.
A1: Correct volume 3600 m3, unit required and 2 s.f.
Note: No Evidence of integration maximum B1 M0A0 M0A0 if a correct answer stated
| Scheme | Marks | AO |
|---|---|---|
| \(350x^2 = (13 + y)^2\left(169 - y^2\right)\) Or \(411x^2 = (13 + y)^2\left(169 - y^2\right)\) | B1 | 3.5c |
| (1) |
Notes
B1: See scheme for equation
| Scheme | Marks | AO |
|---|---|---|
| e.g. the balloon may not be exactly the same shape as the curve The balloon’s material will have some thickness the balloon is not the same shape as the curve as the balloon does not taper to nothing at the bottom. Balloon may stretch B0 for balloon may not be smooth, comments on the basket | B1 | 3.5b |
| (1) | ||
| (8 marks) |
Notes
B1: Correct limitation, see scheme, must be about the balloon part not the basket. Balloon might not be smooth is B0