$($b$)$ The coordinates of two points on the Normal Consolidation Line of a soil are ${}_1^{\text{'}}=$

$400$kPa,$e_1=0\mathrm{.}80$ and ${}_2^{\text{'}}=800$kPa,$e_2=0\mathrm{.}75($where ${}^{\text{'}}$ represents vertical

effective stress and e represents void ratio$).$ In the field, a $3m$ thick normally

consolidated layer of this soil is to be subjected to construction works that will

increase the average vertical stress in the later from $250$kPa to $450$kPa. Determine:

$($i$)$ The initial void ratio before construction. Use a clear labelled graph to explain

your reasoning.

$($ii$)$ The expected $1-$D consolidation settlement as a result of the construction

works.

$($iii$)$ The stress increment that would generate $25mm$ final settlement.

$($iv$)$ The specific volume after the stress increment in $($iii$).$