Math puzzle: The Ant Goes Marching
June the ant has to crawl along the surface of four different 3-D objects to reach her dinner. Your job is to help her find the shortest path along the surface of each object, so that she can chow down as quickly as possible. June is one hungry ant!
1. Suppose June starts at one corner of a cube with a side length of 1 meter. Her dinner is at the opposite corner of the cube. What’s the shortest path she can take to reach her food, traveling along the surface of the cube? What’s the length of this path?
2. Instead of a cube, now suppose June is at one corner of a rectangular prism that is 3 meters long, 2 meters wide and 1 meter tall. Her dinner is at the opposite corner. What’s the shortest path to her food? What’s the length of this path?
3. Now June is on a cylinder. More specifically, she is on the edge of one of the cylinder’s two circular faces. Her dinner is on the edge of the opposite circular face, and all the way around on the other side of that face. The cylinder’s radius is 2 meters and its height is 2 meters. What’s the shortest path to her food? What’s the length of this path?
4. Finally, June is on a hollowed-out cylinder, also known as a “cylindrical shell.” The shell’s outer radius is 2 meters and its inner radius is 1 meter. The shell is 2 meters tall. June is on the outer edge of one of the cylinder’s two flat faces. Her dinner is on the edge of the opposite face, and all the way around on the other end of that face. What’s the shortest path to her food? What’s the length of this path?
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