(i) The altitude of a research drone $H$ , measured in decameters, is modeled by the function

\text{e}^{-0.4t} \sec 0.5t, \quad 0 \le t < \pi

where $t$ is the time in minutes after deployment.

Hence determine the time $t$ at which the altitude of the drone is stationary.

A separate flight path is defined by the implicit relationship

\ln(2\sin y), \quad 0 < y < \frac{\pi}{2}

Show that

\frac{\text{d}y}{\text{d}x} = \frac{\text{e}^x}{\text{f}(x)}

where $f(x)\text{f}(x)$ is a function of $ex\text{e}^x$ that should be found.