Given the schematic illustrations $($Figure $5\mathrm{.}1)$ of the simple cubic $($loosest$)$ packing and tetrahedral $($closest$)$ packing arrangements of idealized, non$-$deformable solid particles of dry sand:

$($A$)$ Verify that the maximum void ratio and porosity of the idealized sandy soil, in its

loosest state, are indeed $0\mathrm{.}91$ and $0\mathrm{.}476(47\mathrm{.}6\%),$ respectively. A single solid particle with radius R would fit a cubical unit volume with dimensions $2$R x $2$R x $2$R $(8$R$^3).($Show all pertinent volume$-$weight relationships and calculations.$)$

$($B$)$ Verify that the minimum void ratio and porosity of the idealized sandy soil, in its

closest state, are indeed $0\mathrm{.}34$ and $0\mathrm{.}26(26\mathrm{.}0\%),$ respectively. In this case, the cubicalunit volume is $5\mathrm{.}657$R$^3,$ versus $8$R$^3$ in simple cubic packing. $($Show all pertinent volume$-$weight relationships and calculations.$)$

$($C$)$ If the specific gravity of the soil solids is $2\mathrm{.}65,$ and water is carefully added so as not to disturb the condition of the sand, show that the saturated water content of the idealized sandy soil, in its loosest state, is $0\mathrm{.}343(34\mathrm{.}3\%).$

$($D$)$ If the specific gravity of the soil solids is $2\mathrm{.}65,$ and water is carefully added so as not to disturb the condition of the sand, show that the saturated water content of the idealized sandy soil, in its closest state, is $0\mathrm{.}128(12\mathrm{.}8\%).$