What is the area of the entire figure (in terms of x) given that the segments AB, CD, EF,…, QR are all semicircle diameters, and the lengths decrease geometrically ( AB=2, CD = 2x, EF= 2x^2, GH = 2x^3,…)?
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What is the area of the entire figure (in terms of x) given that the segments AB, CD, EF,…, QR are all semicircle diameters, and the lengths decrease geometrically ( AB=2, CD = 2x, EF= 2x^2, GH = 2x^3,…)? https://t.co/BxdV2t8447 By @MarshallWBuck pic.twitter.com/DjQURHEYtW
— Mirangu (@Mirangu1) April 26, 2024
One reply on “Cradle of waves”
I counted the area between the pink half-circle and the waves.
Also, I added everything up in the finite geometric series, using the fact that (1-x^2+ x^4-…-x^14) = (1-x^16)/(1+x^2).
I defined c = pi*x^2/2 + arcsin(x) – x*sqrt(1-x^2), which is the area of an “egg-like figure” bounded on one side by a circular arc of radius 1 and on the other side by a semicircle of radius x. Then the full area (that I intended, including the space between the pink semicircle and the wave) is the area of the biggest semicircle, pi/2, minus the egg with area c, plus the egg with area c*x^2, minus the egg with area c*x^4, etc, and finally plus the final egg c*x^14. The final answer is
pi/2 – c*(1-x^16)/(1+x^2).