| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Optimise perimeter or area of 2D region |
| Difficulty | Moderate -0.3 This is a standard C2 optimisation problem with clear scaffolding through parts (a)-(d). Part (a) guides students to derive the area formula from the perimeter constraint, part (b) is routine differentiation and solving dA/dx=0, part (c) uses the second derivative test, and part (d) is simple substitution. While it requires multiple steps, each individual step follows textbook procedures with no novel insight required, making it slightly easier than the average A-level question. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
\includegraphics{figure_3}
Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the stage is $2x$ metres and the width is $y$ metres. The diameter of the semicircular part is $2x$ metres. The perimeter of the stage is 80 m.
\begin{enumerate}[label=(\alph*)]
\item Show that the area, $A$ m², of the stage is given by
$$A = 80x - \left(2 + \frac{\pi}{2}\right)x^2.$$
[4]
\item Use calculus to find the value of $x$ at which $A$ has a stationary value.
[4]
\item Prove that the value of $x$ you found in part (b) gives the maximum value of $A$.
[2]
\item Calculate, to the nearest m², the maximum area of the stage.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q9 [12]}}