Problem $2.(6$ points$)$ A submersible vessel is shaped like a hollow, thin$-$walled sphere. It is made of AISI $1141$ Q&T $315$C $($one of the higher$-$strength alloy steels in the AISI table$).$ Let us assume that manufacturing processes do not permit the shell to be thicker than $100$ mm $.$ What should be the upper limit of the mean diameter for this vessel so that it could be used to dive to the depth of the Titanic wrecks $(3800$ m below sea level$)$ and still maintain a factor of safety of $3$ against yielding? The average specific gravity of ocean water is $1\mathrm{.}025.$ Can the theory of thin$-$wall pressure vessels still be applied to this vessel? Show your work.