$(1)$ A beam with a rectangular cross$-$section shown below is subjected to repeated moment $0M1600lbf.$in$.$ The beam material is ferritic steel with $S_{ut}=60k,Sy=45k,$ if the stress intensity factor is $10k(in{)}^{0\mathrm{.}5}$ and the critical intensity factor is $15$ kpsi $($in$){?}^{0\mathrm{.}5}.$ A crack of size $0\mathrm{.}005$ has been discovered on the bottom of the beam. $($a$)$ Estimate the number of cycles of life remaining for this design. $($b$)$ what is the crack length at a stress equal to the stress intensity factor $(3$ points$)$