An open cylinder garbage can has a height of 4 ft and a circumference of 6 ft. On the inside of the can, 1 ft from the top, is a fly. On the opposite side of the can, 1 ft from the bottom and on the outside, is a lizard. What is the shortest distance that the lizard must walk to reach the fly?
To reach the fly, the lizard must first walk a path on the exterior surface of the garbage can to the opening. From there, it continues walking along a path on the interior surface to catch the prey.
The most salient piece of knowledge we have about minimum distance is that the shortest distance between two points comes from the straight line connecting them. However, it is not immediately apparent how to apply this knowledge in the present problem, since we are constrained to the bent lines on a curved wall. Nonetheless, the curved wall can be cut and flattened, result in a completely flat two dimensional surface. We can then connect the points by a straight line.
Having flattened out the wall in the manner shown in Fig. 1, we see the lizard’s walking path consists two segments of straight line. The length of the path is
Three centuries before the birth of Christ, Euclid discovered the reflection law of light: if a ray of light is reflected by a mirror from point to point , the point of reflection is such that the rays and make equal angles with the mirror ( in Fig. 2). However, it was Heron who first observed that if the point on the mirror were such that , then the resulting total path length would be longer; i.e.,
is minimized when
For the lizard, when , it walks the least (Fig. 1).
From , we have i.e.,
Solving (2) gives
so that (1) yields
Therefore, the shortest distance the lizard must walk is ft.
Exercise-1 Find that minimizes
Exercise-2 If and are real numbers such that , then what is the smallest possible value of
Exercise-3 An open cylinder garbage can has a height of 4 ft and a circumference of 6 ft. On the inside of the can, 1 ft from the top, is a fly. On the opposite side of the can, 1 ft from the bottom inside the can, is a lizard. What is the shortest distance that the lizard must walk to reach the fly?