\begin{enumerate}
  \item \hspace{0pt} [In this question you may assume the following formulae for the volume and curved] surface area of a cone of base radius $r$ and height $h$ and of a sphere of radius $r$.
\end{enumerate}

Cone: volume $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ and curved surface area $S = \pi r \sqrt { h ^ { 2 } + r ^ { 2 } }$\\
Sphere: volume $V = \frac { 4 \pi } { 3 } r ^ { 3 }$ and curved surface area $S = 4 \pi r ^ { 2 }$\\
\includegraphics[max width=\textwidth, alt={}, center]{78ba3acc-4cca-4d15-8362-a27e425c5859-22_782_755_637_657}

Figure 3\\
Figure 3 shows the design for a garden ornament.\\
The ornament is made of a hemisphere on top of a truncated cone.\\
The truncated cone has base radius $2 r \mathrm {~cm}$, top radius $r \mathrm {~cm}$ and height $4 r \mathrm {~cm}$.\\
The hemisphere has radius $R \mathrm {~cm}$.\\
Given that the volume of the ornament is $2100 \pi \mathrm {~cm} ^ { 3 }$\\
(a) show that

$$R ^ { 3 } = 3150 - 14 r ^ { 3 }$$

(b) Find an expression involving $\frac { \mathrm { d } R } { \mathrm {~d} r }$ in terms of $r$ and/or $R$.

The base of the truncated cone of the ornament is fixed to the ground.\\
(c) Show that the visible surface area of the ornament, $A \mathrm {~cm} ^ { 2 }$, is given by

$$A = ( 3 \sqrt { 17 } - 1 ) \pi r ^ { 2 } + 3 \pi R ^ { 2 }$$

(d) Hence show that

$$\frac { \mathrm { d } A } { \mathrm {~d} r } = \gamma \pi r - \frac { \delta \pi r ^ { 2 } } { R }$$

where $\gamma$ and $\delta$ are real numbers to be determined.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-23_705_803_625_630}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

Figure 4 shows a sketch of $A$ against $r$, for $r \geqslant 0$\\
There is a local minimum at $r = 0$ and a local maximum at the point $M$. The overall minimum point is at the point $N$, where the gradient of the curve is undefined.\\
(e) (i) Determine the $r$ coordinate of the point $N$.\\
(ii) Explain why, for the ornament, $r$ must be less than this value.\\
(f) Show that the $r$ coordinate of the point $M$ is

$$\sqrt [ 3 ] { \frac { p ( 3 \sqrt { 17 } - 1 ) ^ { 3 } } { 3 q ^ { 2 } + ( 3 \sqrt { 17 } - 1 ) ^ { 3 } } }$$

where $p$ and $q$ are integers to be determined.

\hfill \mbox{\textit{Edexcel AEA 2023 Q6 [23]}}