5.
\begin{figure}[h]
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\caption{Figure 1}
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An engineering student makes a miniature arch as part of the design for a piece of coursework.
The cross-section of this arch is modelled by the curve with equation
$$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$
where \(a > 1\) and \(A\) is a positive constant. The curve begins and ends on the \(x\)-axis, as shown in Figure 1.
- Show that the length of this curve is \(k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)\), stating the value of the constant \(k\).
The length of the curved cross-section of the miniature arch is required to be 2 m long.
- Find the height of the arch, according to this model, giving your answer to 2 significant figures.
- Find also the width of the base of the arch giving your answer to 2 significant figures.
- Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
(You do not need to consider the arc length of your curve)