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.\\
(i) Show that $a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)$.\\

(ii) Use the equation in part (i) to show by calculation that $1.5 < a < 1.6$.\\

(iii) 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.\\

\hfill \mbox{\textit{CAIE P2 2018 Q6}}