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Two Little Circles

TwoCircles

In the diagram shown we see a big circle of radius R that is tangent to both the x and y axes. What is the radius of the little circle to its southwestern corner? Express its radius in terms of R. (The little circle is also required to be tangent to the axes. The diagram is draw for a circle of radius 2 but we’re working with a general radius R.)

After having solved this problem, the next question will be this:

You can easily see that we can repeat the process by looking at the southwestern circles for ever and ever because there will always be a gap, and you get smaller and smaller circles with small radii. What if you add up all their radii? Do they add up? If so to what number do they add up, in terms of the original radius R. If they don’t add up, why don’t they?

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  1. maffblogger
    June 19, 2015 at 11:36 pm

    Why, that would have to be 0.5*R(sqrt(2)+1)

  2. June 20, 2015 at 10:59 am

    Yes, very good, that would be the answer for the second problem (sum of radii). I have another one coming that’s a little tougher but doable.

  3. maffblogger
    June 20, 2015 at 3:41 pm

    Cool! Well it always good to have your problems doable. I hadn’t given the answer to the first question or any other details for that matter mostly out of not wanting to ruin the surprise for anyone (though if I was serious about that I probably could have kept my mouth shut altogether), but I think you can answer the second question without the first if you’re careful. Anyway, I like your stuff, good work!

  4. June 21, 2015 at 2:40 pm

    Thank you for your kind words and contribution. The answer to the second problem is almost ‘easily’ gotten by inspection since it is related to \sqrt{2}R - R (by looking at the diagonal of the R by R square). 🙂

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