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.
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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.
- 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 \(0 < \alpha < 90\). - Find the value of \(\alpha\) to one decimal place.
Find, according to this model, - the minimum height of the sea bird above sea level, giving your answer to the nearest cm,
[0pt]
[2] - the limitation on the value of \(T\).