$1.(20$ points$)$ A tensile test was performed on a cylindrical specimen having an original diameter of $4$ in $.$ and height of $8$ in $.$ The resulting stress$-$strain curve from the test is shown in the figure. Based on this stress$-$strain curve:

a$)$ Determine approximately the modulus of elasticity, the proportional limit stress, the ultimate stress, and the rupture stress.

b$)$ Determine the rupture strain and the

height of the specimen at the time of rupture.