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Question: An ant is crawling at a rate of one foot per minute along a strip of rubber which can be infinitely and uniformly stretched. The strip is initially one yard long and one inch wide and is stretched an additional yard at the end of each minute. If the ant starts at one end of the strip of rubber, will it ever reach the other end, and if so, when? Answer: We need to find if and when the ant first has less than one foot remaining after the instant when the strip has been stretched. When the strip is stretched, the ants's new position is proportional to its position prior to the stretching. At the end of nth stretching, the distance remaining for the ant to march is (n + 1)/n of the distance remaining immediately before stretching, as the strip has grown to (s + 1) yards from s yards. The rest is an exercise in brute force. Start: 3-1 = 2 feet (the distance remaining prior to the stretch) 2 * 2/1 = 4 feet remaining after the stretch at the end of minute 1. Subtract 1 foot (the distance the ant travels), multiply by 3/2 = 4.5 feet remaining after the stretch at the end of minute 2. Continuing this pattern, subtract 1, multiply by (n + 1)/n, compare the result with 1. The result is first below 1 after the 10th minute, and the distance remaining at that point is 0.78 and change. The ant marches this distance before the next stretching, and therefore reaches the other end of the strip in just over 10.78 minutes. -- Kent Hartman |
