A highway subgrade consists of a 0.5-meter lift of highly plastic clay ($CH$). The project specifications require compaction to 95% of the standard Proctor maximum dry density.

During the construction of an earth dam core, the fill material (a silty clay) is being compacted using a heavy tamping roller. Field density tests reveal that the dry density is consistently falling short of the required 98% of the modified Proctor maximum. The field moisture content is recorded at $12\%$, while the lab optimum moisture content (OMC) is $16\%$.

A dynamic compaction program is being designed for a site with loose, clean sand. The contractor plans to use a tamper weight ($W$) of $15\text{ metric tons}$ dropped from a height ($H$) of $20\text{ meters}$. Assuming an empirical coefficient ($n$) of $0.5$ for this specific granular soil, determine the maximum expected depth of improvement ($D$).

A port facility requires dynamic compaction to treat a $12\text{ meter}$ deep layer of loose silty sand to mitigate liquefaction. The maximum lifting height of the available crane is $25\text{ meters}$. Based on field trials, the empirical coefficient $n$ is determined to be $0.45$. Calculate the minimum tamper weight required to achieve the desired $12\text{ meter}$ depth of improvement.

An engineer is comparing two different dynamic compaction rig configurations to achieve an improvement depth of $10\text{ meters}$ in a heterogeneous landfill material ($n = 0.4$). Rig A offers a $20\text{-ton}$ weight. Rig B offers a $30\text{-ton}$ weight. Calculate the required drop height for both rigs, and determine which rig requires less mechanical lifting energy (potential energy per blow, $E_p = W \cdot H$) to achieve the required depth.

A $200\text{-hectare}$ artificial island is being constructed using hydraulic dredging. The fill material is a clean, uniform, medium-grained silica sand. The sand is deposited in a highly saturated, loose state. The island must support heavy industrial facilities and resist earthquake-induced liquefaction.