$**$Please show all work and explain each step! I am stuck on $5$c & $6$a in particular. Thank you! $**$

Problem $5.$ Consider the design of a storm sewer system. The sewer flow carrying capacity $Q_c(\frac{f{t}^3}{s})$ 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 $6.$ In problem $5,$ 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.