PLEASE COMPLETE ALL PARTS FOR BOTH PROBLEMS!!! $[$Problem $4($a$-$c and Problem $5($a$-$c$)]$

Problem $4.$ Consider the design of a storm sewer system. The sewer flow carrying capacity $Q_c(f\frac{t^3}{s})$ is determined by Manning's equation:

$Q_c=\frac{0\mathrm{.}463}{n}{D}^{\frac{8}{3}}{S}^{\frac{1}{2}}$

where $n$ is Manning's roughness coefficient, $D$ is the pipe

diameter in $ft,$ and $S$ is the pipe slope $).$ The statistical

properties of the flow carrying capacity parameters are

listed in the table. Assume that the system parameters are

independent.

a$)$ Compute the first order approximation of the mean and variance of flow carrying capacity.

b$)$ Compute the second order approximation of the mean of flow carrying capacity.

c$)$ Assuming that the flow carrying capacity is lognormal distributed, compute the probability that it exceeds $30$ cubic feet per second.

Problem $5.$ In problem $4,$ assume that only the pipe diameter $D$ is a random variable and is lognormal distributed with mean $3\mathrm{.}0ft$ and coefficient of variation of $0\mathrm{.}02.$ Assume the other parameters are

constant with $n=0\mathrm{.}015$ and $S=0\mathrm{.}005.$ Using the exact $($analytical$)$ solutions, compute:

a$)$ The mean and variance of flow carrying capacity.

b$)$ The probability distribution of flow carrying capacity.

c$)$ The probability that flow carrying capacity of the system exceeds $30$ cubic feet per second.