A rain barrel, shown in the figure below, is designed to catch rainwater off the shed at a local community garden and provide water at a spigot. The top of the rain barrel is about $12$ ft off the ground and is open to the atmosphere. A PVC pipe with $0\mathrm{.}75-inc{h}_{?}$ inner diameter runs from the base of the rain barrel to a globe valve and spigot located $3$ foff the ground. Several pipe fittings are present in the PVCline and K values for each fitting are given in the figure below. The total pipe length is $20$ fI and the roughness for PVC pipe is $lon=6{10}^{-5}$ inches

If the rain barrel is full $($water surface $12$ ft off the ground$),$ answer the following:

$($a$)$ Assume $f={0\mathrm{.}02}_{?}$ and find the velocity at the exit of the spigot

Velocity $=q,f\frac{t}{s}-1$

$($b$)$ Find the discharge from the spigot in $cfs($assuming $f=0\mathrm{.}02)$

Discharge $=q,$

$($c$)$ Evaluate the initial assumption of $f=0\mathrm{.}02$ and determine your next guess for $f.$ Assume that the water kinematic viscosity is $v=1{10}^{-5}f\frac{t^2}{s}.$

$f=,\frac{3}{3}\mathrm{.}56{10}^{-7-x-1}$

$($d$)$ Assuming the flowrate you calculated in $($b$)$ is correct, how long will it take to fill a $5-$gallon bucket from this rain barrel? $($Note: $1f{t}^3=7\mathrm{.}48$gal $)$

Time $=q,secsx-15$