The problem in the image requires finding the minimum amount of cable needed to connect the nine locations shown in the graph. This is essentially a minimum spanning tree problem, where we need to find the shortest path that connects all nodes (locations) without creating any cycles.

To solve this, we can apply Kruskal's algorithm or Prim's algorithm to find the minimum spanning tree. Both of these algorithms involve selecting edges of the graph in such a way that the total weight (distance in miles) is minimized while connecting all the vertices.

Steps:

1. List all edges with weights:

   • Theater–Home: 4 miles
   • Home–School: 2 miles
   • School–Cleaners: 5 miles
   • Cleaners–Hospital: 6 miles
   • Hospital–Grocery: 4 miles
   • Grocery–Pool: 3 miles
   • Theater–Pool: 2 miles
   • Theater–Library: 7 miles
   • Library–Grocery: 4 miles
   • Big Burger–Hospital: 6 miles

2. Sort edges by increasing weight:

   • Home–School: 2 miles
   • Theater–Pool: 2 miles
   • School–Cleaners: 5 miles
   • Pool–Grocery: 3 miles
   • Hospital–Grocery: 4 miles
   • Library–Grocery: 4 miles
   • Theater–Home: 4 miles
   • Cleaners–Hospital: 6 miles
   • Big Burger–Hospital: 6 miles
   • Theater–Library: 7 miles

3. Use Kruskal's algorithm:

   • Start by picking the smallest edges while ensuring no cycles are formed.
   • The edges chosen will be: Home–School (2 miles), Theater–Pool (2 miles), Pool–Grocery (3 miles), Hospital–Grocery (4 miles), Theater–Home (4 miles), School–Cleaners (5 miles), Cleaners–Hospital (6 miles), Big Burger–Hospital (6 miles).

   Summing these selected edges gives the total length of the minimum spanning tree: $2 + 2 + 3 + 4 + 4 + 5 + 6 + 6 = 32 \text{ miles}.$

Thus, the minimum amount of cable needed is 32 miles.

Let me know if you'd like a deeper explanation of Kruskal's algorithm or further clarification!

Would you like more details on any of the following?

1. How Kruskal's algorithm works.
2. How to identify cycles in graphs.
3. Why this is a minimum spanning tree problem.
4. Alternative methods like Prim's algorithm.
5. Real-world applications of this algorithm.

Tip: When solving minimum spanning tree problems, always make sure to avoid forming cycles while selecting the shortest available edge.