$($a$)(pts)$ Determine the average rainfall excess $($in inces$)$ in the area shown in the sketch for a $60-$min duration storm event with a $10-$yr return period. The CN for the pasture area is $50$ and for the city business area is $95.$ The rainfall intensity formula is given below

Rainfall intensity $=I=\frac{c}{T_d^e+f}($ I is in in $\frac{?}{hr},T_d$ is duration in minutes$)$

where $c=62\mathrm{.}5,e=0\mathrm{.}89$ and $f=9\mathrm{.}10.$

$($b$))$ Determine the lag time $($in min$).$

$($c$)(10pts)$ Determine the diameter of the fully filled culvert at the outlet of the area, given that the culvert's slope is $0\mathrm{.}1\%$ and the Manning's roughness coefficient is $0\mathrm{.}012.$ Unit conversions are provided in the formula sheet. Note; To determine the design flow rate of the culvert, assume a triangular hydrograph.