8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-22_812_958_244_555}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation \(y = x ( x + 2 ) ( x - 4 )\).
The region \(R _ { 1 }\) shown shaded in Figure 2 is bounded by the curve and the negative \(x\)-axis.
- Show that the exact area of \(R _ { 1 }\) is \(\frac { 20 } { 3 }\)
The region \(R _ { 2 }\) also shown shaded in Figure 2 is bounded by the curve, the positive \(x\)-axis and the line with equation \(x = b\), where \(b\) is a positive constant and \(0 < b < 4\)
Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
- verify that \(b\) satisfies the equation
$$( b + 2 ) ^ { 2 } \left( 3 b ^ { 2 } - 20 b + 20 \right) = 0$$
The roots of the equation \(3 b ^ { 2 } - 20 b + 20 = 0\) are 1.225 and 5.442 to 3 decimal places. The value of \(b\) is therefore 1.225 to 3 decimal places.
- Explain, with the aid of a diagram, the significance of the root 5.442