7 A scientist is studying the flight of seabirds in a colony. She models the height above sea level, $H$ metres, of one of the birds in the colony by the equation

$$H = \frac { 140 } { A + 45 \sin 2 t ^ { \circ } - 28 \cos 2 t ^ { \circ } } , \quad 0 \leq t \leq T ,$$

where $t$ seconds is the time after the bird leaves its nest, and $A$ and $T$ are constants.

\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 is a sketch showing the graph of $H$ against $t$.\\
It is given that this seabird's nest is $\mathbf { 2 0 } \mathbf { ~ m }$ above sea level.
\begin{enumerate}[label=(\alph*)]
\item Show that $A = 35$.\\

It is also given that

$$45 \sin 2 t ^ { \circ } - 28 \cos 2 t ^ { \circ } \equiv 53 \sin ( 2 t - \alpha ) ^ { \circ }$$

where $\alpha$ is a constant in the range $0 < \alpha < 90$.
\item Find the value of $\alpha$ to one decimal place.\\

Find, according to this model,
\item the minimum height of the sea bird above sea level, giving your answer to the nearest cm,\\[0pt]
[2]
\item the limitation on the value of $T$.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q7 [8]}}