Example $5$: A circle of diameter $d=9$ in is scribed on an unstressed aluminum plate of thickness $t=0\mathrm{.}75-$in$.$ Forces acting in the plane of the plate later cause normal stress ${}_x=12$ksi and ${}_z=20$ksi. For $E=10{10}^6$ and $v=\frac{1}{3}$ determine the change in: $($a$)$ The length of diameter $AB,($b$)$ the length of diameter $CD,($c$)$ the thickness of the plate and $($d$)$ the volume of the plate.