- During 2029, the number of hours of daylight per day in London, H, is modelled by the equation
$$H = 0.3 \sin \left( \frac { x } { 60 } \right) - 4 \cos \left( \frac { x } { 60 } \right) + 11.5 \quad 0 \leqslant x < 365$$
where \(x\) is the number of days after 1st January 2029 and the angle is in radians.
- Show that, according to the model, the number of hours of daylight in London on the 31st January 2029 will be 8.13 to 3 significant figures.
- Use the substitution \(t = \tan \left( \frac { x } { 120 } \right)\) to show that \(H\) can be written as
$$H = \frac { a t ^ { 2 } + b t + c } { 1 + t ^ { 2 } }$$
where \(a\), \(b\) and \(c\) are constants to be determined.
- Hence determine, according to the model, the date of the first day of 2029 when there will be at least 12 hours of daylight in London.