7 At a town centre car park the length of stay in hours is denoted by the random variable \(X\), which has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 3 } { 2 } } & 1 \leqslant x \leqslant 9
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Interpret the inequalities \(1 \leqslant x \leqslant 9\) in the definition of \(\mathrm { f } ( x )\) in the context of the question.
2. Show that \(k = \frac { 3 } { 4 }\).
3. Calculate the mean length of stay. The charge for a length of stay of \(x\) hours is \(\left( 1 - \mathrm { e } ^ { - x } \right)\) dollars.
4. Find the length of stay for the charge to be at least 0.75 dollars
5. Find the probability of the charge being at least 0.75 dollars.