I thought of a problem a few days ago and I have no idea as to its solution. I posted this on Reddit and xkcd forums earlier but not much has been solved apart from the area of one circle. Suppose you have a boundary formed by the curve y=e^(-x), and the lines x=0 and y=0. In this boundary you place the largest possible circle you can, which touches the y-axis, the x-axis and y=e^(-x). You then place the next largest possible circle to the right of this one, which touches the first circle, y=e^(-x) and the x-axis. This process is repeated indefinitely. If there are an infinite number of circles formed in this way, what is the sum of their areas?(adsbygoogle = window.adsbygoogle || []).push({});

I suspect that the answer will be pretty complicated. Just working out the x-value of the point at which the first circle touches the curve involves working out an x such that x-sqrt((1+e^(2x))x^(2))+sqrt(2xe^(3x))-e^(x)=0, which is a transcendental equation. Someone pointed out that this equation is equivalent to the more compact but still transcendental 4xcosh x = (1+x)^2. As for the remaining infinite circles, I have to idea.

Thanks.

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# Fitting circles under a curve

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