6 The waiting time, \(T\) minutes, for patients at a doctor's surgery has probability density function given by
$$\mathrm { f } ( t ) = \begin{cases} k \left( 225 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 15
0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
- Show that \(k = \frac { 1 } { 2250 }\).
- Find the probability that a patient has to wait for more than 10 minutes.
- Find the mean waiting time.