6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.
- Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
- Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
- Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.