12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-20_775_1015_260_459}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { 3 } { 4 } x ^ { 2 } - 4 \sqrt { x } + 7 , \quad x > 0$$
The point \(P\) lies on \(C\) and has coordinates \(( 4,11 )\).
Line \(l\) is the tangent to \(C\) at the point \(P\).
- Use calculus to show that \(l\) has equation \(y = 5 x - 9\)
The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(x = 1\), the \(x\)-axis and the line \(l\).
- Find, by using calculus, the area of \(R\), giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)