For the constant head permeability test apparatus shown, cross$-$sectional area A $=78\mathrm{.}5c{m}^2$ perpendicular to the direction of flow, distance I between standpipes $=10\mathrm{.}0cm.$ Flow is vertically downwards through the sample. When the head loss $h$ between the two standpipes is $5$ cm $,$ the volume of water which flows through the sample is $100c{m}^3$ per minute.

$($a$)$ What is the hydraulic gradient through the sample?

$($b$)$ What is your estimate of the coefficient of permeability $k$ in $c\frac{m}{sec}?$

$($c$)$ The exit point of the water is maneuvered so that the head loss $h$ is increased $10$ cm $.$ How much water $($in $c{m}^3)$ will flow through the sample in $2$ minutes?