| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Applied context: real-world solid |
| Difficulty | Challenging +1.2 This is a multi-part applied volumes of revolution question requiring: reading off a value from context (a), substituting to find k (b), setting up and evaluating a volume of revolution integral (c), and evaluating the model (d,e). The integration involves rotating y=1+kx² which is standard, but the real-world context and multi-step setup add moderate complexity. Overall a solid but not particularly challenging Core Pure question. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Depth \(= 0.16\) (m) | B1 | Infers that the maximum depth of the bird bath could be \(0.16\) m |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = 1 + kx^2 \Rightarrow 1.16 = 1 + k(0.2)^2 \Rightarrow k = \ldots\) | M1 | Substitutes \(y=1.16\) and \(x=0.2\) or \(x=-0.2\) into \(y=1+kx^2\) and rearranges to give \(k=\ldots\) |
| \(\Rightarrow k = 4\) cao \(\{\text{So } y = 1 + 4x^2\}\) | A1 | \(k=4\) cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{\pi}{4}\int(y-1)\,dy\) or \(\frac{\pi}{4}\int y\,dy\) | B1ft | Uses the model to obtain either \(\frac{\pi}{(\text{their }k)}\int(y-1)\,dy\) or \(\frac{\pi}{(\text{their }k)}\int y\,dy\) |
| \(= \left\{\frac{\pi}{4}\right\}\int_{1}^{1.16}(y-1)\,dy\) or \(= \left\{\frac{\pi}{4}\right\}\int_{0}^{0.16} y\,dy\) | M1 | Chooses limits appropriate to their model |
| \(= \left\{\frac{\pi}{4}\right\}\left[\frac{y^2}{2} - y\right]_{1}^{1.16}\) or \(= \left\{\frac{\pi}{4}\right\}\left[\frac{y^2}{2}\right]_{0}^{0.16}\) | M1, A1 | Integrates \(y\) (w.r.t. \(y\)) to give \(\pm\lambda y^2\); correct integration |
| \(= \frac{\pi}{4}\left(\left(\frac{1.16^2}{2} - 1.16\right) - \left(\frac{1}{2}-1\right)\right) \{= 0.0032\pi\}\) or \(= \frac{\pi}{4}\left(\frac{0.16^2}{2}\right) \{= 0.0032\pi\}\) | — | — |
| \(V_{\text{cylinder}} = \pi(0.2)^2(1.16) \{= 0.0464\pi\}\) | B1 | \(V_{\text{cylinder}} = \pi(0.2)^2(1.16)\) or \(0.0464\pi\) |
| Volume \(= 0.0464\pi - 0.0032\pi \{= 0.0432\pi\}\) | M1 | Depends on both previous M marks; uses model to find \(V_{\text{their cylinder}}\) minus their integrated volume |
| \(= 0.1357\ldots = 0.136\text{ m}^3\) (3sf) | A1 | \(0.136\) cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any one of e.g.: the measurements may not be accurate; the inside surface of the bowl may not be smooth; there may be wastage of concrete when making the bird bath | B1 | States an acceptable limitation of the model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Some comment consistent with their values, e.g. \(\left(\frac{0.136 - 0.127}{0.127}\right)\times 100 = 7.09\ldots\%\); so not a good estimate because the volume of concrete needed is approximately 7% lower than predicted by the model | B1ft | Compares actual volume with their answer to (c); makes an assessment of the model with a reason |
## Question 7:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Depth $= 0.16$ (m) | B1 | Infers that the maximum depth of the bird bath could be $0.16$ m |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = 1 + kx^2 \Rightarrow 1.16 = 1 + k(0.2)^2 \Rightarrow k = \ldots$ | M1 | Substitutes $y=1.16$ and $x=0.2$ or $x=-0.2$ into $y=1+kx^2$ and rearranges to give $k=\ldots$ |
| $\Rightarrow k = 4$ cao $\{\text{So } y = 1 + 4x^2\}$ | A1 | $k=4$ cao |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{\pi}{4}\int(y-1)\,dy$ or $\frac{\pi}{4}\int y\,dy$ | B1ft | Uses the model to obtain either $\frac{\pi}{(\text{their }k)}\int(y-1)\,dy$ or $\frac{\pi}{(\text{their }k)}\int y\,dy$ |
| $= \left\{\frac{\pi}{4}\right\}\int_{1}^{1.16}(y-1)\,dy$ or $= \left\{\frac{\pi}{4}\right\}\int_{0}^{0.16} y\,dy$ | M1 | Chooses limits appropriate to their model |
| $= \left\{\frac{\pi}{4}\right\}\left[\frac{y^2}{2} - y\right]_{1}^{1.16}$ or $= \left\{\frac{\pi}{4}\right\}\left[\frac{y^2}{2}\right]_{0}^{0.16}$ | M1, A1 | Integrates $y$ (w.r.t. $y$) to give $\pm\lambda y^2$; correct integration |
| $= \frac{\pi}{4}\left(\left(\frac{1.16^2}{2} - 1.16\right) - \left(\frac{1}{2}-1\right)\right) \{= 0.0032\pi\}$ or $= \frac{\pi}{4}\left(\frac{0.16^2}{2}\right) \{= 0.0032\pi\}$ | — | — |
| $V_{\text{cylinder}} = \pi(0.2)^2(1.16) \{= 0.0464\pi\}$ | B1 | $V_{\text{cylinder}} = \pi(0.2)^2(1.16)$ or $0.0464\pi$ |
| Volume $= 0.0464\pi - 0.0032\pi \{= 0.0432\pi\}$ | M1 | Depends on both previous M marks; uses model to find $V_{\text{their cylinder}}$ minus their integrated volume |
| $= 0.1357\ldots = 0.136\text{ m}^3$ (3sf) | A1 | $0.136$ cao |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any one of e.g.: the measurements may not be accurate; the inside surface of the bowl may not be smooth; there may be wastage of concrete when making the bird bath | B1 | States an acceptable limitation of the model |
### Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Some comment consistent with their values, e.g. $\left(\frac{0.136 - 0.127}{0.127}\right)\times 100 = 7.09\ldots\%$; so not a good estimate because the volume of concrete needed is approximately 7% lower than predicted by the model | B1ft | Compares actual volume with their answer to (c); makes an assessment of the model with a reason |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{75a62878-dd50-4d52-915a-fe329935d97a-14_577_716_360_296}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{75a62878-dd50-4d52-915a-fe329935d97a-14_630_705_296_1153}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 1 shows the central cross-section $A O B C D$ of a circular bird bath, which is made of concrete. Measurements of the height and diameter of the bird bath, and the depth of the bowl of the bird bath have been taken in order to estimate the amount of concrete that was required to make this bird bath.
Using these measurements, the cross-sectional curve CD, shown in Figure 2, is modelled as a curve with equation
$$y = 1 + k x ^ { 2 } \quad - 0.2 \leqslant x \leqslant 0.2$$
where $k$ is a constant and where $O$ is the fixed origin.\\
The height of the bird bath measured 1.16 m and the diameter, $A B$, of the base of the bird bath measured 0.40 m , as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Suggest the maximum depth of the bird bath.
\item Find the value of $k$.
\item Hence find the volume of concrete that was required to make the bird bath according to this model. Give your answer, in $\mathrm { m } ^ { 3 }$, correct to 3 significant figures.
\item State a limitation of the model.
It was later discovered that the volume of concrete used to make the bird bath was $0.127 \mathrm {~m} ^ { 3 }$ correct to 3 significant figures.
\item Using this information and the answer to part (c), evaluate the model, explaining your reasoning.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS Q7 [12]}}