6 The time, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable \(T\) having the cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0
\frac { t ^ { 3 } } { 216 } & 0 \leqslant t \leqslant 6
1 & t > 6 \end{cases}$$

1. Find, to the nearest day, the time within which 90 per cent of patients will have been given an appointment.
2. Find the probability density function of \(T\) for all values of \(t\).
3. Calculate the mean and the variance of \(T\).
4. Calculate the probability that the time that a patient must wait to be given an appointment is more than one standard deviation above the mean.