https://file.aiquickdraw.com/mathbot/file/1724196171302d9f030gk.jpg

To solve the problem shown in the image, I will first analyze the precedence table and the activity network diagram provided. Let's break it down step by step.

### 1. Analyze the Precedence Table
The precedence table lists the activities, their durations, and the activities that must precede them.

| Activity | Duration | Predecessors |
|----------|----------|--------------|
| A        | 6        | -            |
| B        | 3        | A            |
| C        | 5        | A            |
| D        | 2        | B            |
| E        | 4        | C            |
| F        | 1        | D, E         |

### 2. Construct the Activity Network
The activity network diagram has nodes representing the start and finish of activities, and arrows representing the activities themselves.

### 3. Complete the Table
The table appears to be for determining the earliest start, earliest finish, latest start, latest finish, and slack for each activity. Let's fill in this information:

| Activity | Duration | Earliest Start (ES) | Earliest Finish (EF) | Latest Start (LS) | Latest Finish (LF) | Slack |
|----------|----------|---------------------|----------------------|-------------------|--------------------|-------|
| A        | 6        | 0                   | 6                    | 0                 | 6                  | 0     |
| B        | 3        | 6                   | 9                    | 7                 | 10                 | 1     |
| C        | 5        | 6                   | 11                   | 6                 | 11                 | 0     |
| D        | 2        | 9                   | 11                   | 10                | 12                 | 1     |
| E        | 4        | 11                  | 15                   | 11                | 15                 | 0     |
| F        | 1        | 15                  | 16                   | 15                | 16                 | 0     |

### 4. Determine the Critical Path
The critical path is the longest path through the network, which determines the minimum project completion time. The critical path has zero slack.

Based on the table:
- Path A → B → D → F has a total duration of 16 units.
- Path A → C → E → F also has a duration of 16 units.

Both paths have a duration of 16 units, and both are critical paths.

### Answers:
- **Part a**: The table has been completed.
- **Part b**: The critical paths for the project are:
  - \( A \rightarrow C \rightarrow E \rightarrow F \)
  - \( A \rightarrow B \rightarrow D \rightarrow F \)

Would you like to delve into any particular detail or ask further questions? Here are some related questions you might find interesting:

1. How would the project completion time change if activity B's duration increased by 2 units?
2. What happens to the slack if activity C's duration decreases by 1 unit?
3. Can a project have more than one critical path? What does that imply?
4. How does the introduction of a new activity, say G, between activities D and F, affect the network?
5. Why is it important to identify the critical path in project management?

**Tip**: When analyzing a project, always check for multiple critical paths, as they indicate areas where delays could affect the overall project timeline.

Learn how to solve a project management problem using critical path analysis. This detailed solution includes analyzing the precedence table and activity network diagram to determine critical paths and project completion time. Explore different paths and understand the importance of identifying critical paths in project management.

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