An Ant's path
An ant is crawling along the surface of a cylinder with a diameter and a height. What is the shortest path he can crawl from point to point ? Give your answer to 3 decimal places.
Clarification: point is on the edge of the base and point is on the edge of the top of the cylinder on the side.
The answer is 7.448.
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1 solution
There are two cases and we need to compare which one gives us the minimum value.
Case 1: Ant travels from A along the height and the diameter to reach B. In this case, the distance travelled is 4 + 4 = 8 cm.
Case 2: Ant travels from A along the curved surface to reach B. Since A and B are on the opposite side, we can visualize by cutting open the cylinder.
When you cut a cylinder along its height, it forms a rectangle of breadth h and length 2 π r. If you cut along the plane of A, you will observe that B is at the mid-point of this rectangle, thus forming a right angled triangle. We need to find the length of AB that is a line joining the mid-point of the length of the rectangle from point A.
So using pythagoras theorem, we get AB = ( 4 2 + ( 2 π ) 2 ) = 7.448 cm