Practice Shortest Path Intuition in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The minimum-length route connecting two points, whose form depends on the geometry of the underlying space.
On a flat surface the straight line is always the shortest path between any two points.
Example 1
mediumA river runs along the x-axis. Town A is at (2, 3) and town B is at (8, 5), both north of the river. A pumping station on the river at point P(x, 0) connects to both towns. Find x that minimises the total pipe length AP + PB.
Example 2
hardWhat is the shortest path between two points in the Euclidean plane, and why? Then explain why the shortest path on the surface of a sphere is a great circle arc.
Example 3
easyWhat is the shortest distance from point P(4, 3) to the origin? Justify that the straight line gives the minimum.
Example 4
hardAn ant on the surface of a unit cube wants to travel from vertex A = (0,0,0) to the opposite vertex B = (1,1,1) along the surface. What is the shortest surface path, and what is its length?