$(20$ points$)$ You and a few friends are backcountry camping in Western NC$.$ You plan to trek $8$ miles and then set up camp. When setting up camp you and your friends will break into teams to do tasks in unison to set up camp faster. However, some tasks are dependent on others and cannot be completed before the prior tasks have been performed. The itinerary for camping is as follows:

You and your friends hike a total of $8$ miles to the nearest campsite $($Activity A$).$ Upon arriving you break off into teams. One team will set up tents and hammocks for everyone to sleep in $($Activity B$).$ At the same time, another team will get wood for a fire $($Activity C$).$ Once they have the wood, that same team then proceeds to start the fire $($Activity D$).$ The third team will walk to find water $($Activity E$)$ and once they find it they will to collect water and bring it back to camp $($Activity F$).$ Once the fire is ready and the water is brought back to camp the two groups $-$ the one collecting firewood and the one collecting water$-$ will then purify the water $($Activity G$).$ Once the camp is set up and the water is purified everyone will come together and cook dinner around the campfire $($Activity H$),$ which will mark the end of the day.

a$.(9$ points$)$ Assuming that the PERT chart is deterministic $($i$.$e$.,$ based on most likely event times$),$ complete a PERT chart model for the camping trip by determining and placing activity $($label with the letter$),$ activity times, the earliest start times, and the latest start times on the model.

b$.(2$ points$)$ Determine the critical path of the PERT chart in a$.$

c$.(2$ points$)$ Assuming a deterministic system, how long will it take to complete the entire process?

d$.(7$ points$)$ Your friends ask you what time you all should start the hike if they want to be in finished with dinner by $8$pm$.$ What time do you tell them? Note: You can assume they want to be $95\%$ confident in their departure time. $($PERT$-3$ analysis$)$