$6-$ Before loading, the plate shown below is inscribed with a line at a slope of $1$:$2$ relative to

the x $-$axis. The plate is then subjected to the plane$-$stress state shown. Stress in the z $-$

direction $($perpendicular to the page$)$ is zero. Assume Young's Modulus $E=$

$170,000$MPa, shear modulus $G=52,600$MPa, Poisson's ratio $v=0\mathrm{.}33,$ and a yield

stress of $150$ MPa $.($a$)$ Under the given stress state, determine the new slope of the inscribed

line. $($b$)$ Assuming the thickness $($dimension perpendicular to the page$)$ of the plate before

loading was $25$ mm $,$ what is the thickness of the plate under the given stress state?

$[2$ Marks$]$