| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2021 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Applied context: real-world solid |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question with standard techniques. Part (a) requires simple substitution of coordinates into an equation. Part (b) involves routine integration of polynomial and square root terms to find volume using π∫x²dy. Parts (c) and (d) are basic modelling commentary requiring minimal mathematical work. The algebraic manipulation is standard for AS-level, with no novel problem-solving required. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((5,15) \Rightarrow 15 = \frac{\sqrt{225 \times 5^2 - 2025}}{a} \Rightarrow a = \ldots\) | M1 | Substitutes \((5,15)\) into the equation modelling the curve to find \(a\) |
| \(a = 4\) | A1 | Infers from data the value of \(a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evidence of use of \(\pi \int x^2 \, dy\) for curve \(BC\) or curve \(CD\) | M1 | Uses either model to obtain \(x^2\) in terms of \(y\) and applies \(\pi \int x^2 \, dy\) |
| For \(BC\): \(V_1 = \frac{\pi}{225}\int(16y^2 + 2025)\,dy\) or \(\pi\int\left(\frac{16}{225}y^2+9\right)dy\) | A1ft | Correct expression for volume from curve \(BC\) (follow through on \(a\)) |
| For \(CD\): \(V_2 = 25\pi\int(16-y)\,dy\) or \(\pi\int(400-25y)\,dy\) | A1 | Correct expression for volume from curve \(CD\) |
| \(V_1 = \frac{\pi}{225}\int_0^{15}(16y^2+2025)\,dy\) | M1 | Chooses limits appropriate to model for curve \(BC\) |
| \(V_2 = 25\pi\int_{15}^{16}(16-y)\,dy\) | M1 | Chooses limits appropriate to model for curve \(CD\) |
| \(V_1 = \frac{\{\pi\}}{225}\left[\frac{16y^3}{3}+2025y\right]_0^{15}\) | A1ft | Correct integration (follow through on \(a\)) |
| \(V_2 = 25\{\pi\}\left[16y - \frac{y^2}{2}\right]_{15}^{16}\) | A1ft | Correct integration |
| \(V = V_1 + V_2 = \frac{\pi}{225}(18000+30375) + 25\pi\left(128 - \frac{255}{2}\right)\) | M1 | Uses model to find sum of volumes |
| \(V = \frac{455\pi}{2}\ \text{cm}^3\) or \(227.5\pi\ \text{cm}^3\) | A1 | Correct final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| E.g. The equation of the curve may not be a suitable model; the sides will not be perfectly curved/smooth; there will be a hole in the middle for the wick | B1 | States an acceptable limitation of the model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Makes appropriate comment consistent with their volume value and \(700\ \text{cm}^3\). E.g. a good estimate as \(700\ \text{cm}^3\) is only \(15\ \text{cm}^3\) less than \(715\ \text{cm}^3\) | B1ft | Compares actual volume to answer from (b) and makes assessment of the model with a reason |
# Question 9:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(5,15) \Rightarrow 15 = \frac{\sqrt{225 \times 5^2 - 2025}}{a} \Rightarrow a = \ldots$ | M1 | Substitutes $(5,15)$ into the equation modelling the curve to find $a$ |
| $a = 4$ | A1 | Infers from data the value of $a$ |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evidence of use of $\pi \int x^2 \, dy$ for curve $BC$ or curve $CD$ | M1 | Uses either model to obtain $x^2$ in terms of $y$ and applies $\pi \int x^2 \, dy$ |
| For $BC$: $V_1 = \frac{\pi}{225}\int(16y^2 + 2025)\,dy$ or $\pi\int\left(\frac{16}{225}y^2+9\right)dy$ | A1ft | Correct expression for volume from curve $BC$ (follow through on $a$) |
| For $CD$: $V_2 = 25\pi\int(16-y)\,dy$ or $\pi\int(400-25y)\,dy$ | A1 | Correct expression for volume from curve $CD$ |
| $V_1 = \frac{\pi}{225}\int_0^{15}(16y^2+2025)\,dy$ | M1 | Chooses limits appropriate to model for curve $BC$ |
| $V_2 = 25\pi\int_{15}^{16}(16-y)\,dy$ | M1 | Chooses limits appropriate to model for curve $CD$ |
| $V_1 = \frac{\{\pi\}}{225}\left[\frac{16y^3}{3}+2025y\right]_0^{15}$ | A1ft | Correct integration (follow through on $a$) |
| $V_2 = 25\{\pi\}\left[16y - \frac{y^2}{2}\right]_{15}^{16}$ | A1ft | Correct integration |
| $V = V_1 + V_2 = \frac{\pi}{225}(18000+30375) + 25\pi\left(128 - \frac{255}{2}\right)$ | M1 | Uses model to find sum of volumes |
| $V = \frac{455\pi}{2}\ \text{cm}^3$ or $227.5\pi\ \text{cm}^3$ | A1 | Correct final answer |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| E.g. The equation of the curve may not be a suitable model; the sides will not be perfectly curved/smooth; there will be a hole in the middle for the wick | B1 | States an acceptable limitation of the model |
## Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Makes appropriate comment consistent with their volume value and $700\ \text{cm}^3$. E.g. a good estimate as $700\ \text{cm}^3$ is only $15\ \text{cm}^3$ less than $715\ \text{cm}^3$ | B1ft | Compares actual volume to answer from (b) and makes assessment of the model with a reason |9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-28_639_517_255_774}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the vertical cross-section, $A O B C D E$, through the centre of a wax candle.\\
In a model, the candle is formed by rotating the region bounded by the $y$-axis, the line $O B$, the curve $B C$, and the curve $C D$ through $360 ^ { \circ }$ about the $y$-axis.
The point $B$ has coordinates $( 3,0 )$ and the point $C$ has coordinates $( 5,15 )$.\\
The units are in centimetres.\\
The curve $B C$ is represented by the equation
$$y = \frac { \sqrt { 225 x ^ { 2 } - 2025 } } { a } \quad 3 \leqslant x < 5$$
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $a$ according to this model.
The curve $C D$ is represented by the equation
$$y = 16 - 0.04 x ^ { 2 } \quad 0 \leqslant x < 5$$
\item Using algebraic integration, determine, according to the model, the exact volume of wax that would be required to make the candle.
\item State a limitation of the model.
When the candle was manufactured, $700 \mathrm {~cm} ^ { 3 }$ of wax were required.
\item Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2021 Q9 [13]}}