Guided tours of a museum begin every 60 minutes. A randomly chosen tourist arrives \(X\) minutes after the start of a tour. The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { ( x - 20 ) ^ { 2 } } { 24000 } & 0 < x < 60
0 & \text { otherwise } \end{cases}$$ The random variable \(T\) is the time that the tourist has to wait for the next tour to begin. Show that the distribution function G of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 0 & t \leqslant 0
\frac { 8 } { 9 } - \frac { ( 40 - t ) ^ { 3 } } { 72000 } & 0 < t < 60
1 & t \geqslant 60 \end{cases}$$ Find the median and the mean of \(T\).