$(4)$ A minimum and maximum density test was conducted on poorly graded sand using a cylinder

of $5000c{m}^3.$ For the maximum void ratio, dry sand is poured into the container as loose as possible and

weighed. The weight of the dry sand for the maximum void ratio is $7\mathrm{.}27kg.$ In contrast, sand is poured into

the same container and vibrated in fully saturated conditions to determine the minimum void ratio. After the

vibration, the weight of the sand and water is $11\mathrm{.}24kg.$ Knowing that $G_s=2\mathrm{.}67$ calculate:

$($a$)$ The maximum void ratio.

$($b$)$ The minimum void ratio.

$($c$)$ What is the relative density of the soil in situ if the void ratio is $e=0\mathrm{.}63?$

$($d$)$ What are the maximum and minimum dry densities of the sand?

$($e$)$ If the soil needs to be densified to at least $D_r=85\%,$ what is the void ratio and porosity of the sand at

this state?