$(15$ pts$)$ A $5-$foot inside diameter corrugated metal pipe $($CMP$)$ under a bury highway is corroded and needs replacement. The CMP has an estimated Manning's n of $0\mathrm{.}024$ and is constructed at a $\backslash (0\mathrm{.}5\backslash \%\backslash )$ slope. The highway agency's design requirement is for flow depth in culverts not to exceed $0\mathrm{.}7$ times the pipe diameter $(\backslash ($ y $/$ D$=0\mathrm{.}7\backslash )).$

a$.$ Assuming the pipe flows at normal depth, using Manning's equation, what is the existing pipes discharge capacity $($Q$)?$

b$.$ One alternative for rehabilitation is to slipline the existing pipe $($slide another pipe inside the existing$)$ with a $4-$foot diameter high density polyethylene $($HDPE$)$ pipe with an $\backslash (\backslash$mathrm$\{$n$\}=0\mathrm{.}011\backslash ).$ Using the same $\backslash (\backslash$mathrm$\{$y$\}/\backslash$mathrm$\{$D$\}=0\mathrm{.}7\backslash )$ design criteria, what is the discharge capacity of the proposed HDPE slipliner pipe? What percentage of the existing pipes capacity would this solution provide?