2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-04_506_590_255_429}
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\caption{Figure 1}
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\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-04_296_327_456_1311}
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\caption{Figure 2}
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A uniform plane figure \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = \ln 5\), the curve with equation \(y = 8 \mathrm { e } ^ { - x }\) and the line with equation \(x = \ln 2\). The unit of length on each axis is one metre.
The area of \(R\) is \(2.4 \mathrm {~m} ^ { 2 }\)
The centre of mass of \(R\) is at the point with coordinates \(( \bar { x } , \bar { y } )\).
- Use algebraic integration to show that \(\bar { y } = 1.4\)
Figure 2 shows a uniform lamina \(A B C D\), which is the same size and shape as \(R\). The lamina is freely suspended from \(C\) and hangs in equilibrium with \(C B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
- Find the value of \(\theta\)