Standard +0.3 This is a standard A-level optimization problem involving surface area constraint, volume formula derivation, and differentiation to find maximum. Part (a) requires careful geometry (Pythagoras) and algebraic manipulation but follows a predictable pattern. Parts (b)-(d) are routine calculus applications. The multi-step nature and algebraic complexity elevate it slightly above average, but it requires no novel insight—just systematic application of standard techniques.
\includegraphics{figure_2}
The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(ABCDEF\). The base \(ACFD\) of the roof is a horizontal rectangle, and the cross-section \(ABC\) of the roof is an isosceles triangle with \(AB = BC\).
The lengths of \(AC\) and \(CF\) are \(2x\) cm and \(y\) cm respectively, and the height of \(BE\) above the base of the roof is \(x\) cm.
The total surface area of the five faces of the roof is \(600\) cm\(^2\) and the volume of the roof is \(V\) cm\(^3\).
Show that \(V = kx (300 - x^2)\), where \(k = \sqrt{a + b}\) and \(a\) and \(b\) are integers to be determined. [6]
Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum. [4]
Find the maximum volume of the roof. Give your answer in cm\(^3\), correct to the nearest integer. [1]
Explain why, for this roof, \(x\) must be less than a certain value, which you should state. [2]
\begin{enumerate}[label=(\alph*)]
\item
\includegraphics{figure_2}
The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism $ABCDEF$. The base $ACFD$ of the roof is a horizontal rectangle, and the cross-section $ABC$ of the roof is an isosceles triangle with $AB = BC$.
The lengths of $AC$ and $CF$ are $2x$ cm and $y$ cm respectively, and the height of $BE$ above the base of the roof is $x$ cm.
The total surface area of the five faces of the roof is $600$ cm$^2$ and the volume of the roof is $V$ cm$^3$.
Show that $V = kx (300 - x^2)$, where $k = \sqrt{a + b}$ and $a$ and $b$ are integers to be determined. [6]
\item Use differentiation to determine the value of $x$ for which the volume of the roof is a maximum. [4]
\item Find the maximum volume of the roof. Give your answer in cm$^3$, correct to the nearest integer. [1]
\item Explain why, for this roof, $x$ must be less than a certain value, which you should state. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q2 [13]}}