6. The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0
\frac { 1 } { 135 } \left( 54 t + 9 t ^ { 2 } - 4 t ^ { 3 } \right) , & 0 \leq t \leq 3
1 , & t > 3 \end{cases}$$

1. Find the probability that a patient waits for more than 20 minutes.
2. Show that the median waiting time is between 11 and 12 minutes.
3. Define fully the probability density function \(\mathrm { f } ( t )\) of \(T\).
4. Find the modal waiting time in minutes.
5. Give one reason why this model may need to be refined.