SPS SPS SM Pure 2024 June — Question 5 8 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2024
SessionJune
Marks8
TopicStationary points and optimisation
TypeShow formula then optimise: cylinder/prism (single variable)
DifficultyStandard +0.3 This is a standard optimization problem requiring volume constraint substitution, differentiation of a simple rational function, and second derivative test. All steps are routine A-level techniques with no novel insight required, making it slightly easier than average.
Spec1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx
5. In this question you must show all stages of your working. Solutions based entirely on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-10_629_988_370_577} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\), length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)
  1. Show that the surface area of the brick, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
  2. Hence find the value of \(x\) for which \(S\) is stationary and justify that this value of \(x\) gives the minimum value of \(S\).
  3. Hence find the minimum surface area of the brick.
5. In this question you must show all stages of your working. Solutions based entirely on calculator technology are not acceptable.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-10_629_988_370_577}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

A brick is in the shape of a cuboid with width $x \mathrm {~cm}$, length $3 x \mathrm {~cm}$ and height $h \mathrm {~cm}$, as shown in Figure 2.

The volume of the brick is $972 \mathrm {~cm} ^ { 3 }$
\begin{enumerate}[label=(\alph*)]
\item Show that the surface area of the brick, $S \mathrm {~cm} ^ { 2 }$, is given by

$$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
\item Hence find the value of $x$ for which $S$ is stationary and justify that this value of $x$ gives the minimum value of $S$.
\item Hence find the minimum surface area of the brick.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q5 [8]}}
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