## Question 9(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(4,14),(1,18)\Rightarrow 14=a(4)^2+b,\ 18=a(1)^2+b \Rightarrow a=\ldots, b=\ldots$ | M1 | |
| $a=-\dfrac{4}{15},\ b=\dfrac{274}{15}$ | A1 | |

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## Question 9(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\pi\times4^2\times14$ and $\pi\times1^2\times10$ | B1 | |
| $\pi\int x^2\,dy=\dfrac{\pi}{4}\int(274-15y)\,dy$ | B1ft | |
| $=\dfrac{\pi}{4}\displaystyle\int_{14}^{18}(274-15y)\,dy$ | M1 | |
| $=\dfrac{\pi}{4}\left[274y-\dfrac{15y^2}{2}\right]_{14}^{18}$ | M1, A1 | |
| $V=234\pi+\dfrac{\pi}{4}\left[274(18)-\dfrac{15(18)^2}{2}-\left(274(14)-\dfrac{15(14)^2}{2}\right)\right]$ | ddM1 | |
| $V=268\pi\approx842\text{ cm}^3$ | A1 | |

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## Question 9(c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Any one of: measurements may not be accurate; equation of curve may not be suitable model; bottom of bottle may not be flat; thickness of glass not considered; glass may not be smooth | B1 | Must refer to limitation of the model, e.g. measuring of dimensions, model used for curve, simplified model (thickness of glass, shape, smoothness) |

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## Question 9(d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Comparison of their value to 750 (e.g. $810-750=\ldots$) AND conclusion consistent with their value (e.g. not a good model / this is a good model) | B1ft | Must demonstrate comparison, not just state percentage error or difference with no calculation. If answer $<750$ conclude not good model; if $>750$ give sensible consistent comment |

# Question (from mark scheme notes):

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $(4, 14)$ and $(1, 18)$ into the curve equation to obtain at least one correct equation | M1 | Attempts to find values of $a$ and $b$ |
| Correct values of $a$ and $b$ inferred from the data in the model | A1 | |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct expressions for the 2 cylindrical parts | B1 | May be seen as a sum or as separate cylinders |
| Uses the model to obtain $\pi\int\left(\dfrac{y - b}{a}\right)dy$ | B1ft | Note that the $\pi$ may be recovered later |
| Chooses limits appropriate to the model: $14$ and $18$ | M1 | |
| Integrates to obtain an expression of the form $\alpha y + \beta y^2$ | M1 | |
| Uses their model correctly to give $274y - \dfrac{15y^2}{2}$ | A1 | |
| Uses the model to find the sum of their cylinders + their integrated volume | ddM1 | Must be fully correct method; dependent on both previous M marks; must have attempted volumes of cylinders "AHBG" and "CFED" and adds these to the magnitude of their integrated volume |
| $268\pi$ or awrt $842$ | A1 | |

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| States an acceptable limitation of the model with no contradictory statements | B1 | Independent of part (b) |

## Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Compares the actual volume to their answer to part (b) and makes an assessment of the model with a reason | B1ft | No contradictory statements |