7. The fraction of sky covered by cloud is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 0
\mathrm { f } ( x ) = k x ^ { 2 } ( 1 - x ) & 0 \leq x \leq 1 ,
\mathrm { f } ( x ) = 0 & x > 1 . \end{array}$$

1. Find \(k\) and sketch the graph of \(\mathrm { f } ( x )\).
2. Find the mean and the variance of \(X\).
3. Find the cumulative distribution function \(\mathrm { F } ( x )\).
4. Given that flying is prohibited when \(85 \%\) of the sky is covered by cloud, show that cloud conditions allow flying nearly \(90 \%\) of the time.