$[15$ points$]$ In a laboratory falling head test, a sample of soil $8\mathrm{.}75$ cm in diameter and $15$ cm long. It was observed that the head was initially $50$ cm and was reduced by $5$ cm in a minute.

a$.$ If the radius of the standpipe $=10mm,$ calculate the hydraulic conductivity, in $c\frac{m}{s}.$

b$.$ How many minutes is required to the head to fall to $30$ cm $?$

c$.$ If the objective is to decrease the head to $20$ cm within $5$ minutes, determine the appropriate diameter of the standpipe needed, in mm $.$