Problem $4.[20$ pts$]$ Consider a water distribution system consisting of a storage tank serving as the source and a $2-$ft diameter cast$-$iron pipe $1$ mile long, leading to a user. The head water elevation at the

source is maintained at a constant height of $100ft$ above the

levation at the user end. Because of the uncertainty in the

pipe roughness, the supply to the user is not certain. The flow

rate $($in cubic feet per second$)$ in the pipe may be obtained by

the Hazen$-$Williams equation:

$Q_s=\frac{C}{149\mathrm{.}2}(\frac{h}{L}{)}^{0\mathrm{.}54}{D}^{2\mathrm{.}63}$

where $h$ is the head difference $($in feet$)$ between the source and the user, $D$ is the pipe diameter in feet, and $L$ is the pipe length in feet. The roughness $(C)$ of the distribution pipe is a lognormal distributed random variable with a mean of $130$ and a standard deviation of $20.$ The demand on flow rate is $1-$parameter exponential distributed with a mean of $0\mathrm{.}3cfs.$

a$)$ Determine the distribution of system capacity $Q_s.$

b$)$ Compute the system reliability using the direct integration method.