a$)$ Based on the given information regarding the activities for the project, the project completion date $=$ weeks.

b$)$ To reduce the duration of the project to $10$ weeks at the least cost, the following is the order of what needs to be followed while crashing the activities:

The first activity to crash is

by $2$ weeks at a per week crashing cost of dollars.

Next, one should crash activity $D$ and activity

by $2$ weeks for an additional total per week crashing cost of

dollars.

c$)$ New requirements are to reduce the project to $7$ weeks. Continuing to crash the activities from $10$ weeks based on the above crashing already completed, the following is the order for crashing:

Next activity to crash in order is

by $2$ weeks at a per week crashing cost of dollars.

Next, one should crash activity

and activity $E$ by $1$ week at a total per week crashing cost of

dollars.

Total cost of crashing the project to $7$ weeks : $]$ dollars.