$2-$ For the following pipe system:

a$)$ What are the number of nodes and independent paths$/$loops$?(3$ points$)$

b$)$ Derive the governing equations of the flow rates within the pipes. $(10$ points$)$

c$)$ Use hardy$-$cross method in the loop to find the second approximation of $Q3,Q4,$ and Q$5,$ when the initial guess is $Q{3}_i=0\mathrm{.}5f\frac{t^3}{s},Q{i}_i=1f\frac{t^3}{s},$ and $Q{5}_i=1f\frac{t^3}{s}.(2$ points$)$

All pipes have a diameter of $D=0\mathrm{.}2ft,$ length of $l=500ft,$ and friction factor of $f=0\mathrm{.}025.$

Ignore minor losses.