To determine the total duration of the project, we use the Precedence Diagramming Method (PDM) to calculate the Critical Path. The Critical Path is the longest sequence of activities that dictates the minimum time required to complete the project.

Step 1: Map the Dependencies

Activity A and B start simultaneously ($T=0$).

Activity C is a " sink " for A and B. It cannot start until both are finished.

Activity D starts after C is completed.

Step 2: Calculate the Paths

We have two possible paths from the start of the project to the end:

Path 1: A $\rightarrow$ C $\rightarrow$ D

Duration: $1 \text{ (A)} + 2 \text{ (C)} + 5 \text{ (D)} = 8 \text{ weeks}$.

Path 2: B $\rightarrow$ C $\rightarrow$ D

Duration: $3 \text{ (B)} + 2 \text{ (C)} + 5 \text{ (D)} = 10 \text{ weeks}$.

Step 3: Identify the Project Duration

Because Activity C requires both A and B to be finished, it must wait for the longer of the two.

Activity A finishes at end of Week 1.

Activity B finishes at end of Week 3.

Therefore, Activity C starts at the beginning of Week 4.

Calculation:

End of B = Week 3

End of C = $3 \text{ (Start)} + 2 \text{ (Duration)} = \text{Week 5}$

End of D = $5 \text{ (Start)} + 5 \text{ (Duration)} = \text{Week 10}$

The project will take 10 weeks to complete. Path 2 (B-C-D) is the Critical Path.

Analysis of Other Options:

A. Eleven weeks: This would be the result if A and B were sequential rather than parallel ($1+3+2+5=11$).

B. Nine weeks: This does not align with any logical combination of the given activity durations.

C. Eight weeks: This is the duration of the shorter path (A-C-D). However, the project cannot finish until the longest path is completed.