Problem $6(20$ points$)$

The aggregate mix used for the design of an asphalt concrete mixture consists of $41$ percent coarse aggregates, $52$ percent fine aggregates, and $7$ percent mineral filler. If the respective bulk specific gravities of these materials are $2\mathrm{.}603,2\mathrm{.}714,$ and $2\mathrm{.}691,$ and the effective specific gravity of the aggregates is $2\mathrm{.}821,$ determine the optimum asphalt content if results obtained using the Marshall method are shown in the following table. The specific gravity of the asphalt is $1\mathrm{.}02.($Note: use Asphalt Institute method and NAPA method to determine optimum asphalt content$).$

$\backslash$table$[[$Percent$,$Weight of Specimen $($g$),$Stability,Flow$],[$Asphalt$,$In air $($SSD$),$In air $($Dry$),$Under water,$(lb),$ inch$)],[5\mathrm{.}5,1346\mathrm{.}8,1331\mathrm{.}9,789\mathrm{.}5,1805,13],[6\mathrm{.}0,1351\mathrm{.}5,1336\mathrm{.}8,797\mathrm{.}3,1845,14],[6\mathrm{.}5,1358\mathrm{.}5,1342\mathrm{.}9,804\mathrm{.}8,1870,16],[7\mathrm{.}0,1364\mathrm{.}3,1348\mathrm{.}7,808\mathrm{.}5,1827,20],[7\mathrm{.}5,1370\mathrm{.}1,1354\mathrm{.}2,809\mathrm{.}1,1710,25]]$