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03 November 2024

Comments

Michael R. Kesti

Do I correctly understand that R is not the only available data because it is also specified that L is equal to the perimeter length of the small semicircle?

George Rebane

MichaelK 250pm - Good clarifying question. R is indeed the only *data* that is provided. The rest describes the constraining geometry of the problem, such as r and the line's length L which must be computed from the solution once given the numerical value of R.

The Estonian Fox

Let's take the minimum & maximum values of r to see if there is a solution.
rmin = 0, in which case L=2R. Perimeter P = 0.
Let rmax = R (a coincident semicircle to the green one), then L=0. P = (2+pi)*R (Add the semicircle + line segment AD.) Both P and L are continuous functions, so they will intersect if you plot them together. So there is a solution.

Look at the diagram. Angle ABE is 90-deg; angle ACD is also 90-deg, since it is inscribed and intersects the diameter. So these 2 triangles are similar right triangles. So we want to solve for AC (which is L) in the equation (AD/AE) = (AC/AB).

Let r = k*R. Solve ABE rt triangle for AB. AE= R(2-k), EB=k*R, AB^2=[R(2-k)]^2 - k^2R^2.

AB=2R√(1-k); by inspection, AD = 2R; AE = R(2-k). AC=(AD*AB)/AE.

So AC = L= 2R* 2R√(1-k) /(R(2-k)) = 4R*√(1-k)/(2-k). Need to set this equal to k*R(2+pi) (the perimeter).
So 4*√(1-k) / (2-k) = k(2+pi).
Way too complicated for an analytical solution, so I used Excel.
k = 0.37826, then r = 0.37826R and L = 1.9448R

George Rebane

EFox -323pm - Correctamundo! And your use of Excel's gradient search Solver was also the most efficient way to go. Analytically solving the resulting quartic polynomial is like getting into a land war in Asia.

The Estonian Fox

Actually George, I did it the old-fashioned way. Stepped k-column from 0.1 to 0.5 by 0.01. Narrowed easily to 0.38, stepped 0.0001 to get to 4 places, then I eye-balled the final 5th-place digit. Took less than a minute. Since I don't use Solver much, it would have taken me longer to remember how to use it. So, efficient enough for me.

I still use Excel 2007, because I have the ACTUAL DISK for Office 2007. I don't need no stinkin' cloud storage or yearly fee. Office 2007 has been re-loaded at least 7 additional times onto my & my daughter's new & used computers over the years.

The Estonian Fox

And besides, what country would be stupid enough to get into a land war in Asia?

OK, OK, more than twice?

George Rebane

Efox 540am - What country? How about the US? Let us count the ways - Spanish-American (1898+), anti-Bolshevik expedition (Arkhangelsk in September 1918), WW2, Korea, Vietnam, Iraq, Afghanistan,... ;-)

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