https://file.aiquickdraw.com/mathbot/file/1736678071949hhgqp51g.docx

Let's solve **Question 4 and Question 5** based on the details in your document.

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### **Question #4:**
#### Steps to Solve:
1. **Calculate Expected Activity Times (Te):**
   Use the formula for expected time:
   \[
   Te = \frac{\text{Optimistic Time (O)} + 4 \times \text{Most Likely Time (M)} + \text{Pessimistic Time (P)}}{6}
   \]
2. **Calculate Variance (\(V\)):**
   Use the formula for variance:
   \[
   V = \left(\frac{\text{Pessimistic Time (P)} - \text{Optimistic Time (O)}}{6}\right)^2
   \]
3. **Identify Critical Path:**
   Using the precedence relationships, construct the network diagram, determine the path(s), and calculate the total duration for each path to find the critical path.
4. **Probability of Completing in 50 Days:**
   Use the z-score formula:
   \[
   Z = \frac{\text{Target Time (T) - Expected Project Time (Te)}}{\sqrt{\text{Critical Path Variance (Sum of V along the path)}}}
   \]

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### **Question #5:**
#### Steps to Solve:
1. **Identify Critical Path for the Normal Schedule:**
   - Calculate the total duration of each path in the network using the normal times.
   - Identify the longest path as the critical path.
2. **Crash Analysis:**
   - Calculate the cost slope for each activity:
     \[
     \text{Cost Slope} = \frac{\text{Crash Cost - Normal Cost}}{\text{Normal Time - Crash Time}}
     \]
   - Prioritize crashing activities on the critical path with the smallest cost slope until the project is reduced by two days.

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Would you like me to solve these explicitly with detailed calculations for expected time, variance, and critical path for **both questions**, or should I start with one and proceed step by step?

Learn how to solve questions on the critical path, calculate expected time and variance, and apply probability in project management. This guide covers project scheduling, cost slope analysis, and variance calculations for advanced problem-solving in project management, suitable for college students and professionals.

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