1. **Problem Statement:** Find the minimal spanning tree (MST) and the associated shortest total distance for the given weighted network of city nodes. 2. **Formula and Concept:** The minimal spanning tree of a weighted graph is a subset of edges that connects all vertices with the minimum possible total edge weight and no cycles. 3. **Method:** We use Kruskal's algorithm, which sorts edges by weight and adds the smallest edge that does not form a cycle until all nodes are connected. 4. **Edges sorted by weight:** - Blackpool–Preston 17 - Carlisle–Penrith 19 - Kendal–Lancaster 20 - Preston–Liverpool 30 - Leeds–York 24 - Scotch Corner–Middlesbrough 26 - Penrith–Kendal 28 - Skipton–Leeds 28 - Workington–Carlisle 32 - Manchester–Liverpool 35 - Preston–Skipton 35 - Newcastle upon Tyne–Middlesbrough 34 - Kendal–Barrow-in-Furness 34 - Leeds–Sheffield 33 - Carlisle–Newcastle upon Tyne 57 - Workington–Penrith 37 - Manchester–Sheffield 38 - Scotch Corner–Newcastle upon Tyne 41 - Lancaster–Skipton 42 - Penrith–Scotch Corner 50 - Middlesbrough–York 50 - Workington–Barrow-in-Furness 59 - Leeds–Manchester 40 - Scotch Corner–Leeds 54 5. **Step-by-step MST construction:** - Add Blackpool–Preston (17) - Add Carlisle–Penrith (19) - Add Kendal–Lancaster (20) - Add Leeds–York (24) - Add Scotch Corner–Middlesbrough (26) - Add Penrith–Kendal (28) - Add Skipton–Leeds (28) - Add Workington–Carlisle (32) - Add Newcastle upon Tyne–Middlesbrough (34) - Add Kendal–Barrow-in-Furness (34) - Add Leeds–Sheffield (33) - Add Preston–Liverpool (30) - Add Preston–Skipton (35) - Add Manchester–Liverpool (35) 6. **Check for cycles and connectivity:** All nodes are connected without cycles. 7. **Calculate total distance:** $$17 + 19 + 20 + 24 + 26 + 28 + 28 + 32 + 34 + 34 + 33 + 30 + 35 + 35 = 395$$ **Final answer:** The minimal spanning tree has a total shortest distance of **395** units.