| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| 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 optimisation problem with clear scaffolding through four parts. Students must form a constraint equation from surface area, substitute to get V(r), differentiate a polynomial, and verify a maximum using the second derivative test. All techniques are routine for this level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | 6 | 2 |
Question 9:
9 | 6 | 2 | 2 | 4 | 2 | 1 | 11A pencil holder is in the shape of an open circular cylinder of radius $r$ cm and height $h$ cm. The surface area of the cylinder (including the base) is 250 cm$^2$.
\begin{enumerate}[label=(\alph*)]
\item Show that the volume, V cm$^3$, of the cylinder is given by $V = 125r - \frac{\pi r^3}{2}$. [4]
\item Use calculus to find the value of $r$ for which $V$ has a stationary value. [3]
\item Prove that the value of $r$ you found in part (b) gives a maximum value for $V$. [2]
\item Calculate, to the nearest cm$^3$, the maximum volume of the pencil holder. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q9 [11]}}