| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and related rates |
| Difficulty | Standard +0.8 This question combines volumes of revolution about the y-axis (requiring rearrangement and integration by parts) with related rates of change. Part (i) requires careful setup of the volume integral and non-trivial integration involving exponentials. Part (ii) applies implicit differentiation to connect rates. While systematic, it demands multiple advanced techniques and careful algebraic manipulation beyond standard textbook exercises. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to express \(x\) in terms of \(y\) | *M1 | obtaining two terms |
| Obtain \(x = e^{\frac{y}{2}} + 1\) | A1 | or equiv |
| State or imply volume involves \(\int\pi x^2\) | B1 | |
| Attempt to express \(x^2\) in terms of \(y\) | *M1 | dep *M; expanding to produce at least 3 terms |
| Obtain \(k\int(e^y + 2e^{\frac{y}{2}} + 1)dy\) | A1 | any constant \(k\) including 1; allow if \(dy\) absent |
| Integrate to obtain \(k(e^y + 4e^{\frac{y}{2}} + y)\) | A1 | |
| Use limits 0 and \(p\) | M1 | dep *M *M; evidence of use of 0 needed |
| Obtain \(\pi(e^p + 4e^{\frac{1}{2}p} + p - 5)\) | A1 8 | AG; necessary detail required |
| (ii) State or imply \(\frac{dp}{dr} = 0.2\) | B1 | maybe implied by use of 0.2 in product |
| Obtain \(\pi(e^p + 2e^{\frac{1}{2}p} + 1)\) as derivative of \(V\) | B1 | |
| Attempt multiplication of values or expressions for \(\frac{dp}{dr}\) and \(\frac{dV}{dp}\) | M1 | |
| Obtain \(0.2\pi(e^p + 2e^2 + 1)\) | A1√ | following their \(\frac{dV}{dp}\) expression |
| Obtain \(44\) | A1 5 | or greater accuracy |
**(i)** Attempt to express $x$ in terms of $y$ | *M1 | obtaining two terms
Obtain $x = e^{\frac{y}{2}} + 1$ | A1 | or equiv
State or imply volume involves $\int\pi x^2$ | B1 |
Attempt to express $x^2$ in terms of $y$ | *M1 | dep *M; expanding to produce at least 3 terms
Obtain $k\int(e^y + 2e^{\frac{y}{2}} + 1)dy$ | A1 | any constant $k$ including 1; allow if $dy$ absent
Integrate to obtain $k(e^y + 4e^{\frac{y}{2}} + y)$ | A1 |
Use limits 0 and $p$ | M1 | dep *M *M; evidence of use of 0 needed
Obtain $\pi(e^p + 4e^{\frac{1}{2}p} + p - 5)$ | A1 8 | AG; necessary detail required
**(ii)** State or imply $\frac{dp}{dr} = 0.2$ | B1 | maybe implied by use of 0.2 in product
Obtain $\pi(e^p + 2e^{\frac{1}{2}p} + 1)$ as derivative of $V$ | B1 |
Attempt multiplication of values or expressions for $\frac{dp}{dr}$ and $\frac{dV}{dp}$ | M1 |
Obtain $0.2\pi(e^p + 2e^2 + 1)$ | A1√ | following their $\frac{dV}{dp}$ expression
Obtain $44$ | A1 5 | or greater accuracy9\\
\includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-4_556_720_676_715}
The diagram shows the curve with equation $y = 2 \ln ( x - 1 )$. The point $P$ has coordinates ( $0 , p$ ). The region $R$, shaded in the diagram, is bounded by the curve and the lines $x = 0 , y = 0$ and $y = p$. The units on the axes are centimetres. The region $R$ is rotated completely about the $\boldsymbol { y }$-axis to form a solid.\\
(i) Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the solid is given by
$$V = \pi \left( \mathrm { e } ^ { p } + 4 \mathrm { e } ^ { \frac { 1 } { 2 } p } + p - 5 \right) .$$
(ii) It is given that the point $P$ is moving in the positive direction along the $y$-axis at a constant rate of $0.2 \mathrm {~cm} \mathrm {~min} ^ { - 1 }$. Find the rate at which the volume of the solid is increasing at the instant when $p = 4$, giving your answer correct to 2 significant figures.
\hfill \mbox{\textit{OCR C3 2006 Q9 [13]}}