1. On a particular day, the depth of water in a river estuary at a specific location is modelled by the equation

$$D = 2 \sin \left( \frac { x } { 3 } \right) + 3 \cos \left( \frac { x } { 3 } \right) + 6 \quad 0 \leqslant x \leqslant 7 \pi$$ where the depth of water is \(D\) metres at time \(x\) hours after midnight on that day.

1. Write down the depth of water at midnight, according to the model. Using the substitution \(t = \tan \left( \frac { x } { 6 } \right)\)
2. show that equation (I) can be re-written as $$D = \frac { 3 t ^ { 2 } + 4 t + 9 } { 1 + t ^ { 2 } }$$
3. Hence determine, according to the model, the time after midnight when the depth of water is 5 metres for the first time. Give your answer to the nearest minute.