| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Show formula then optimise: cylinder/prism (single variable) |
| Difficulty | Standard +0.3 This is a standard C2 optimization problem with a volume constraint. Part (a) is straightforward substitution using V=πr²h=500. Parts (b-d) involve routine differentiation, solving dA/dr=0, and applying the second derivative test—all textbook techniques with no novel insight required. Slightly easier than average due to clear structure and standard methods. |
| Spec | 1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = \pi r^2 h = 500\), \(A = 2\pi r h + \pi r^2\) | B1, M1 | |
| \(A = 2\pi r\left(\frac{500}{\pi r^2}\right) + \pi r^2 = \pi r^2 + \frac{1000}{r}\) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dA}{dr} = 2\pi r - 1000r^{-2}\) | M1 A1 | |
| \(2\pi r - 1000r^{-2} = 0\), \(r = \sqrt[3]{\frac{500}{\pi}}\) (\(\approx 5.42\)) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d^2A}{dr^2} = 2\pi + 2000r^{-3} > 0\), therefore minimum | M1 A1 ft | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = \pi r^2 + \frac{1000}{r} = 277\) (nearest integer) | M1 A1 | (2 marks) |
# Question 6:
## Part (a):
$V = \pi r^2 h = 500$, $A = 2\pi r h + \pi r^2$ | B1, M1 |
$A = 2\pi r\left(\frac{500}{\pi r^2}\right) + \pi r^2 = \pi r^2 + \frac{1000}{r}$ | M1 A1 | (4 marks)
## Part (b):
$\frac{dA}{dr} = 2\pi r - 1000r^{-2}$ | M1 A1 |
$2\pi r - 1000r^{-2} = 0$, $r = \sqrt[3]{\frac{500}{\pi}}$ ($\approx 5.42$) | M1 A1 | (4 marks)
## Part (c):
$\frac{d^2A}{dr^2} = 2\pi + 2000r^{-3} > 0$, therefore minimum | M1 A1 ft | (2 marks)
## Part (d):
$A = \pi r^2 + \frac{1000}{r} = 277$ (nearest integer) | M1 A1 | (2 marks)
---6. A container made from thin metal is in the shape of a right circular cylinder with height $h \mathrm {~cm}$ and base radius $r \mathrm {~cm}$. The container has no lid. When full of water, the container holds $500 \mathrm {~cm} ^ { 3 }$ of water.
\begin{enumerate}[label=(\alph*)]
\item Show that the exterior surface area, $A \mathrm {~cm} ^ { 2 }$, of the container is given by $A = \pi r ^ { 2 } + \frac { 1000 } { r }$.
\item Find the value of $r$ for which $A$ is a minimum.
\item Prove that this value of $r$ gives a minimum value of $A$.
\item Calculate the minimum value of $A$, giving your answer to the nearest integer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [12]}}