|Optimal Loading of Camels|
This activity introduces exponential decrease in a surprising way.
Suppose we want to transport thousands of bananas from a banana plantation to a destination 400 miles away, using camels to transport the bananas. Each camel can carry a maximum load of 400 bananas, and each camel consumes one banana for each mile it travels towards the destination. Clearly, if each camel is loaded with 400 bananas there will be no bananas left when the camel reaches its destination 400 miles away.
So, we decide to use the camels more efficiently. Start with an even number of camels, each carrying a maximum load of 400 bananas, and group them in pairs (so each pair carries 800 bananas) and have them carry the bananas 200 miles to the halfway point. Each camel consumes 200 bananas on the way, so each pair consumes 400 bananas and delivers 400 to the halfway point. Then we reload these 400 bananas on one of the camels of each pair and let him carry the full load the rest of the way to the destination. He consumes 200 bananas on the way, and 200 bananas survive the trip for each pair we started with. So, by using a second station for reloading we have increased the survival rate from 0 to 200/800 = 0.25, or 25%.
This works so well, we decide to use teams of 64 camels, each team carrying a total load of 64 times 400, or 25,600 bananas. Each team carries the bananas 100 miles to the one-quarter point, where there is a reloading station. Each camel consumes 100 bananas on the way, so each team consumes 6400 bananas and the number surviving is 25,600 - 6400 = 19,200. We divide these bananas among a new team of 48 camels (giving each camel the maximum load of 400 bananas) and have them carry their loads another 100 miles to the next reloading station at the halfway point. Each of the 48 camels consumes 100 bananas, so the number surviving to the halfway point is 19,200 - 4800 = 14,400 bananas. We divide these among 36 camels (giving each the maximum load of 400 bananas) and let them carry their loads another 100 miles to the next reloading station at the three-quarters point. They consume 3600 bananas on the way, leaving us with 14,400 - 3600 = 10,800 bananas. We divide these among 27 camels (giving each the maximum load of 400 bananas) and they carry their loads the final 100 miles, consuming 2700 bananas on the way and delivering 10,800 - 2700 = 8100 bananas to the final destination. In this way, the survival rate is 8100/25,600 = 0.3164, or 31.64%.
If we repeat the process, using more and more camels and more and more loading stations, an easy calculation shows that the survival rate for n equally spaced loading stations is equal to (1 - 1/n)n. As n increases without bound this survival rate approaches the limiting value 1/e = 0.3679, or 36.79%. An applet draws a graph showing the number of bananas surviving at each reloading station. At present, the viewer can click on the position for locating each reloading station.