$3.$ Force and length change data are given in the table below for the initial portion of a tension test on AISI $4140$ steel tempered at $\backslash (538^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The diameter before testing was $10$ mm $,$ and the gage length $\backslash ($ L$\_\{$i$\}\backslash )$ for the length change measurement was $50$ mm $.$

$($a$)$ Calculate corresponding values of engineering stress and strain, and plot the engineering stress$-$engineering strain curve. Use Excel or a similar tool to generate the plot.

$($b$)$ Compute and plot true stress vs$.$ true strain. Plot on the same graph as engineering stress vs$.$ engineering strain so that you can clearly see the differences between the curves. Hand$-$drawn plots will NOT be accepted.

$($c$)$ Determine the $\backslash (0\mathrm{.}2\backslash \%\backslash )$ offset yield strength.

$($d$)$ What tensile load is required to cause yielding in a bar of the same material but with a diameter of $25$ mm $?$ How does this value compare with the load at yielding in the $10$ mm diameter test specimen? Why do the two values differ?