Towns A to F are to be connected by cable along existing roads. Need to figure the minimum length of cable required. Eash town must be connected to at least one other town. Distances are in miles. A-B 20, A-E 71, A-F 38, B-A 20, B-C 58, B-E 62,C-B 58, C-D 17, C-E 42, D-C 17, D-E 40, E-A 71, E-B 62, E-C 42, E-F 52, F-A 38, F-E 52 I say A-B 20, C-D 17, E-F 52 for 89 miles?

Question

Towns A to F are to be connected by cable along existing roads. Need to figure the minimum length of cable required. Eash town must be connected to at least one other town. Distances are in miles. A-B 20, A-E 71, A-F 38, B-A 20, B-C 58, B-E 62,C-B 58, C-D 17, C-E 42, D-C 17, D-E 40, E-A 71, E-B 62, E-C 42, E-F 52, F-A 38, F-E 52 I say A-B 20, C-D 17, E-F 52 for 89 miles?

Answer 1

To find the minimum length of cable required to connect all towns A to F, you can use an algorithm called "Minimum Spanning Tree" (MST). One well-known algorithm to find MST is Kruskal's algorithm. Here's how you can apply it to this scenario:

1. List all the distances between the towns provided:

A-B 20, A-E 71, A-F 38
B-C 58, B-E 62
C-D 17, C-E 42
D-E 40
E-F 52

2. Sort the distances in ascending order:

C-D 17
A-B 20
A-F 38
C-E 42
D-E 40
E-F 52
B-C 58
B-E 62
A-E 71

3. Start with an empty edge list and an empty set of included towns.

4. Iterate through the sorted distances and add each distance (edge) to the edge list, checking if it connects two towns that are not already included.

5. Stop when all towns are connected (i.e., when the number of edges in the edge list is equal to the number of towns - 1).

6. The sum of the distances in the edge list is the minimum length of cable required.

Applying this algorithm to the provided distances, here's the step-by-step process:

1. Start with an empty edge list and an empty set of included towns.

2. Add the first distance C-D 17 to the edge list, connecting C and D.

3. Add the second distance A-B 20 to the edge list, connecting A and B.

4. Add the third distance A-F 38 to the edge list, connecting A and F.

5. Add the fourth distance C-E 42 to the edge list, connecting C and E.

6. Since we have included all towns, stop the process.

The resulting edge list is: C-D 17, A-B 20, A-F 38, C-E 42

7. Calculate the sum of the distances in the edge list: 17 + 20 + 38 + 42 = 117 miles.

Therefore, the answer is that the minimum length of cable required to connect all towns A to F is 117 miles, not 89 miles as you suggested.