| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Optimization with constraints |
| Difficulty | Standard +0.3 This is a standard C2 optimization problem with clear scaffolding through five parts. Students must form an expression from volume constraint, derive surface area formula (given answer provided), differentiate, solve a cubic equation (which simplifies nicely to x³=257.5), and verify minimum using second derivative. While it requires multiple techniques, each step is routine and the question structure guides students through the process. Slightly easier than average due to heavy scaffolding and standard methods. |
| 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 |
\includegraphics{figure_3}
A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions $2x$ cm by $x$ cm and height $h$ cm, as shown in Fig. 4.
Given that the capacity of a carton has to be 1030 cm$^3$,
\begin{enumerate}[label=(\alph*)]
\item express $h$ in terms of $x$, [2]
\item show that the surface area, $A$ cm$^2$, of a carton is given by
$$A = 4x^2 + \frac{3090}{x}.$$ [3]
\end{enumerate}
The manufacturer needs to minimise the surface area of a carton.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use calculus to find the value of $x$ for which $A$ is a minimum. [5]
\item Calculate the minimum value of $A$. [2]
\item Prove that this value of $A$ is a minimum. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [14]}}