Water Droplet Rings

Algebra Level 5

After a hard rainfall, Bobby decides to go outside to go out to the river behind his house to watch raindrops drip down from the trees above. He notices that a particular tree has one drop fall into the river every second. When the drop hits the water it creates a ring that grows continuously outward at a rate of \(π \text{ cm}^2 \) per second. As more drops fall, more rings are created in concentric circles around the drop point. Because the Fibonacci Sequence often shows up in nature, after watching many drops fall and seeing many water rings created, Bobby wonders if there's ever a point in time where there are 6 rings that differ in radius by the first 5 terms of the Fibonacci sequence. An example of what Bobby is thinking of can be seen below.

Please note that these rings are not consecutive as shown in the photo; there are other rings of varying radii in between each of the 6 rings.

Bobby soon realizes that there are infinitely many points in time where this can occur, so naturally the question arises: at what point in time does this first occur? If this time can be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

Details and Assumptions:

  • Assume \(t=0\) as the point in which the first drop hits the water.

  • Note that the growth rate is in terms of the area of the circle, not the radius.

  • Circles of radius 0 are not considered circles.


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