7 The queuing time, \(T\) minutes, for a person queuing at a supermarket checkout has probability density function given by
$$f ( t ) = \begin{cases} c t \left( 25 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 5
0 & \text { otherwise } \end{cases}$$
where \(c\) is a constant.
- Show that the value of \(c\) is \(\frac { 4 } { 625 }\).
- Find the probability that a person will have to queue for between 2 and 4 minutes.
- Find the mean queuing time.