Fill the ocean
Imagine an empty giant swimming pool whose volume is the same as the Pacific Ocean's volume. Someone suddenly opens a tap that starts filling the pool with water while a chronometer starts running. During the first minute, 10 gallons of water are released. At the end of the second minute, there are 15 gallons of water inside the pool. And when the chronometer shows that 6 minutes have passed, there are 24.5 gallons of water inside the pool. If the water continues following this pattern forever, is it possible to determine if the pool will be ever completely filled by the water?
Consider that:
- No water evaporates.
- It is possible to have an amount of water smaller than an atom.
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1 solution
If you have been studying sums of infinite divergent series, you may have easily figured out that the pattern followed by the water is the harmonic series multiplied by ten. The series is defined by 10/n where n is the number of minutes that have passed and "10/n" is the amount of water flowing each n minute. The series begins 10/1, 10/2, 10/3, 10/4, 10/5, 10/6 and goes on infinitely.
You may have noticed that after 1 minute, the tap has released only 10 gallons of water, exactly as the problem states.
After 2 minutes, there have been released 10/1 + 10/2 gallons of water, which is 10 + 5 = 15 gallons of water.
After 6 minutes: 10/1 + 10/2 + 10/3 + 10/4 + 10/5 + 10/6 = 10 + 5 + 20/6 + 2.5 + 2 + 10/6 = 15 + 4.5 + 30/6 = 19.5 + 5 = 24.5 gallons of water.
It is known that the sum of a harmonic series is infinite, a feature that can be easily proved by the comparison test, integral test, and other tests. It means that a water flow which follows this pattern forever tends to have an infinite amount of water. Therefore, the pool will be completely filled by water if the water keeps this pattern forever, exactly as the problem states, independently of its volume.