6 A design for a surfboard is shown in Figure 1.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-06_415_1403_447_322}
\end{figure}
The curve of the top half of the surfboard can be modelled by the parametric equations
$$\begin{aligned}
& x = - 2 t ^ { 2 } \\
& y = 9 t - 0.7 t ^ { 2 }
\end{aligned}$$
for \(0 \leq t \leq 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-06_383_1342_1379_351}
\end{figure}
6
- Find the length of the surfboard.
6 - Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
6
- (ii) Hence, show that the width of the surfboard is approximately one third of its length.
\(7 \quad\) A planet takes \(T\) days to complete one orbit of the Sun.
\(T\) is known to be related to the planet's average distance \(d\), in millions of kilometres, from the Sun.
A graph of \(\log _ { 10 } T\) against \(\log _ { 10 } d\) is shown with data for Mercury and Uranus labelled.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-08_752_1447_580_296}