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To find the shortest path from vertex \(E\) to vertex \(F\) in the given graph, we can apply Dijkstra's algorithm or simply inspect the paths manually since the graph is small.

### Step 1: List all possible paths from \(E\) to \(F\):
1. \(E \rightarrow A \rightarrow C \rightarrow F\)
2. \(E \rightarrow A \rightarrow D \rightarrow F\)
3. \(E \rightarrow B \rightarrow D \rightarrow F\)
4. \(E \rightarrow B \rightarrow C \rightarrow F\)
5. \(E \rightarrow D \rightarrow F\)

### Step 2: Calculate the total weights for each path:
1. \(E \rightarrow A \rightarrow C \rightarrow F: 3 + 6 + 6 = 15\)
2. \(E \rightarrow A \rightarrow D \rightarrow F: 3 + 1 + 3 = 7\)
3. \(E \rightarrow B \rightarrow D \rightarrow F: 7 + 4 + 3 = 14\)
4. \(E \rightarrow B \rightarrow C \rightarrow F: 7 + 1 + 6 = 14\)
5. \(E \rightarrow D \rightarrow F: 7 + 3 = 10\)

### Step 3: Identify the shortest path:
- The shortest path is \(E \rightarrow A \rightarrow D \rightarrow F\) with a total weight of 7.

Thus, the shortest path from \(E\) to \(F\) is **\(E \rightarrow A \rightarrow D \rightarrow F\)** with a distance of 7.

Would you like to explore more details or have any further questions?

### Here are some related questions:
1. What is Dijkstra's algorithm, and how does it work?
2. How would the shortest path change if the weight between \(E\) and \(B\) was reduced?
3. Can you explain why some paths are longer than others?
4. How does this problem relate to real-world applications of shortest path algorithms?
5. What if we were looking for the longest path instead of the shortest?

**Tip:** When solving shortest path problems, ensure you evaluate all possible paths to avoid missing the optimal solution.

Find the shortest path from E to F in the given graph.

Learn how to find the shortest path from E to F in a graph. We calculate all possible paths and apply Dijkstra's algorithm to determine that the shortest path is E → A → D → F with a total distance of 7. Suitable for students in Grades 10-12 studying graph theory.

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