5 A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-05_453_1200_358_242} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations \(x = 1 + 0.2 t - \cos t , \quad y = k \sin ^ { 2 } t\), where \(k\) is a positive constant and \(0 \leqslant t \leqslant \pi\). Lengths are in metres and the area of the emblem must be \(1 \mathrm {~m} ^ { 2 }\).

1. Show that \(k \int _ { 0 } ^ { \pi } \left( 0.2 + \sin t - 0.2 \cos ^ { 2 } t - \sin t \cos ^ { 2 } t \right) \mathrm { d } t = 1\).
2. Determine the exact value of \(k\).