| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 12 |
| Topic | Tangents, normals and gradients |
| Type | Optimization with constraints |
| Difficulty | Standard +0.3 This is a standard optimization problem requiring volume formula for a prism with sector cross-section, surface area calculation, and basic calculus (differentiation to find minimum). The algebra is straightforward, and the method is a routine textbook exercise in applied calculus, making it slightly easier than average for A-level. |
| Spec | 1.02z Models in context: use functions in modelling1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
\includegraphics{figure_10}
The figure above shows solid right prism of height $h$ cm.
The cross section of the prism is a circular sector of radius $r$ cm, subtending an angle of 2 radians at the centre.
\begin{enumerate}[label=(\alph*)]
\item Given that the volume of the prism is 1000 cm$^3$, show clearly that
$$S = 2r^2 + \frac{4000}{r},$$
where $S$ cm$^2$ is the total surface area of the prism. [5]
\item Hence determine the value of $r$ and the value of $h$ which make $S$ least, fully justifying your answer. [7]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q10 [12]}}