9 A water slide is the shape of a curve \(P Q\) as shown in Figure 1 below.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-14_592_1278_427_470}
\end{figure}
The curve can be modelled by the parametric equations
$$\begin{aligned}
& x = t - \frac { 1 } { t } + 4.8
& y = t + \frac { 2 } { t }
\end{aligned}$$
where \(0.2 \leq t \leq 3\)
The horizontal distance from \(O\) is \(x\) metres.
The vertical distance above the point \(O\) at ground level is \(y\) metres.
\(P\) is the point where \(t = 0.2\) and \(Q\) is the point where \(t = 3\)
9
- To make sure speeds are safe at \(Q\), the difference in height between \(P\) and \(Q\) must be less than 7 metres.
Show that the slide meets this safety requirement.
9 - Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\)
9
- (ii) A vertical support, \(R S\), is to be added between the ground and the lowest point on the slide as shown in Figure 2 below.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-16_590_1278_475_470}
\end{figure}
Find the length of \(R S\)
9 - (iii) Find the acute angle the slide makes with the horizontal at \(Q\)
Give your answer to the nearest degree.
\section*{END OF SECTION A}