3 For a restaurant with a home-delivery service, the delivery time in minutes can be modelled by a continuous random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { \pi } { 90 } \sin \left( \frac { \pi t } { 60 } \right) & 20 \leqslant t \leqslant 60
0 & \text { otherwise. } \end{cases}$$

1. Given that \(20 \leqslant a \leqslant 60\), show that \(\mathrm { P } ( T \leqslant a ) = \frac { 1 } { 3 } \left( 1 - 2 \cos \left( \frac { \pi a } { 60 } \right) \right)\). There is a delivery charge of \(\pounds 3\) but this is reduced to \(\pounds 2\) if the delivery time exceeds a minutes.
2. Find the value of \(a\) for which the expected value of the delivery charge for a home-delivery is £2.80.