The value
4.47204 from Geometers' Sketchpad could be
R*2*sqrt(5). (That identification of the number 4.47204 comes from A Dictionary of Real Numbers, by Jonathan Borwein and Peter Borwein, Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, California, 1990.)
Via calculus we want to maximize 2*sin(x) + 4*cos(x), leaving R constant of course:
Its derivative is 2*cos(x)  4*sin(x)
Equate this to zero so that
cos(x)=2*sin(x)
tan(x)=1/2
x=26.5650511771 degrees
2*sin(x)+4*cos(x)=4.472135955
That value squared = 20, whose square root is indeed 2*sqrt(5).
So the maximum perimeter is in fact R*2*sqrt(5).

Posted by Charlie
on 20171107 10:07:58 