$(1)$ A saturated clay specimen is tested in the Oedometer $($consolidation cell$).$ The initial diameter and height of the specimen are $75mm$ and $20mm,$ respectively. The initial water content was $62\%$ and $G_s=2\mathrm{.}7.$ The height of the solids $H_5$ is $10mm.$ The vertical stresses were applied incrementally, meaning each increment remains on the specimen until the pore water pressure change was negligible. The cumulative change in height values at the end of each consolidation step are as follows: $(10$ points$)$

$\backslash$table$[[$Vertical stress : p$($kPa$),15,30,60,120,240,480],[$Cumulative Change in Height : $\backslash$Delta H$($mm$),0\mathrm{.}10,0\mathrm{.}11,0\mathrm{.}21,1\mathrm{.}13,2\mathrm{.}17,3\mathrm{.}15],[$Final Specimen Height : H$\_(($l$))($mm$),,,,,,],[$Height of Voids : H$\_($v$)($mm$),,,,,,],[$Final Void Ratio : e$,,,,,,]]$

$($a$)$ Calculate the compression index for pressure $60$kPa using t$90$ method

$($b$)$ Calculate the $C_c$

Final Void Ratio $($e$)$