$2.$ A student wants to develop a project using PERT to modify one of products which has poor sales. She has determined the following activities, precedence relationships and time estimates in days for the project:

TIME estimate Exp Time Variance

Act to tm tp te Vt

a $4675\mathrm{.}830\mathrm{.}250$

b $1232\mathrm{.}000\mathrm{.}111$

c $5555\mathrm{.}000\mathrm{.}000$

d $79129\mathrm{.}170\mathrm{.}694$

e $2484\mathrm{.}331\mathrm{.}000$

f $69108\mathrm{.}670\mathrm{.}444$

a$)$ Name all potential paths and corresponding durations

b$)$ Calculate ES$,$ EF$,$ LS$,$ LF$,$ and slack for all activities.

c$)$ How many days can activity $"$e$"$ be delayed without extending project duration?