7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-6_559_923_292_670}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

Figure 4 shows a shape $S ( \theta )$ made up of five line segments $A B , B C , C D , D E$ and $E A$ . The lengths of the sides are $A B = B C = 5 \mathrm {~cm} , C D = E A = 3 \mathrm {~cm}$ and $D E = 7 \mathrm {~cm}$ . Angle $B A E =$ angle $B C D = \theta$ radians.

The length of each line segment always remains the same but the value of $\theta$ can be varied so that different symmetrical shapes can be formed,with the added restriction that none of the line segments cross.
\begin{enumerate}[label=(\alph*)]
\item Sketch $S ( \pi )$ ,labelling the vertices clearly.

The shape $S ( \phi )$ is a trapezium.
\item Sketch $S ( \phi )$ and calculate the value of $\phi$ .

The smallest possible value for $\theta$ is $\alpha$ ,where $\alpha > 0$ ,and the largest possible value for $\theta$ is $\beta$ , where $\beta > \pi$ .
\item Show that $\alpha = \arccos \left( \frac { 29 } { 40 } \right) \cdot \left[ \arccos ( x ) \right.$ is an alternative notation for $\left. \cos ^ { - 1 } ( x ) \right]$
\item Find the value of $\beta$ .

The area,in $\mathrm { cm } ^ { 2 }$ ,of shape $S ( \theta )$ is $R ( \theta )$ .
\item Show that for $\alpha \leqslant \theta < \pi$

$$R ( \theta ) = 15 \sin \theta + \frac { 7 } { 4 } \sqrt { 87 - 120 \cos \theta }$$

Given that this formula for $R ( \theta )$ holds for $\alpha \leqslant \theta \leqslant \beta$
\item show that $R ( \theta )$ has only one stationary point and that this occurs when $\theta = \frac { 2 \pi } { 3 }$
\item find the maximum and minimum values of $R ( \theta )$.

FOR STYLE, CLARITY AND PRESENTATION: 7 MARKS TOTAL FOR PAPER: 100 MARKS\\
END
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2018 Q7 [27]}}