Solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$
given that \(x = 20\) when \(t = \frac { \pi } { 4 }\), giving your solution in the form \(x ^ { 2 } = \mathrm { f } ( t )\). (6 marks)
The oscillations of a 'baby bouncy cradle' are modelled by the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$
where \(x \mathrm {~cm}\) is the height of the cradle above its base \(t\) seconds after the cradle begins to oscillate.
Given that the cradle is 20 cm above its base at time \(t = \frac { \pi } { 4 }\) seconds, find:
the height of the cradle above its base 13 seconds after it starts oscillating, giving your answer to the nearest millimetre;
the time at which the cradle will first be 11 cm above its base, giving your answer to the nearest tenth of a second.
(2 marks)