circles problem work

Problem statement
=================

Imagine a red circle.

Around the red circle, imagine a set of identical yellow circles; each
yellow circle is tangent to the red circle and to each of its
neighbors, like balls in a ring bearing.

Around the yellow circles, imagine a surrounding circle tangent to
each of the yellow circles (so this circle and the red circle are
concentric). Any area inside the outer circle that's not already red
or yellow, color it blue.

Let the radius of each yellow circle be r, and the radius of the red
circle be R.

What ratio of r to R maximizes the blue area? The number of yellow
circles can vary as needed, as long as the other qualities are met (no
gaps between yellow circles, no partial circles).

==============================

      c
      *
     /|\
    /x|x\
   /  R  \
  /   |   \
 /    |    \
*-----*-----*
   r     r

let R = 1
let r > 0 in Reals
let x > 0, x < 2*pi in Reals
let y > 1, in Integers
let red_area, yellow_area, and blue_area > 0 in Reals

tan(x) = r
x = arctan(r) = (2*pi*2) / y
> x is some whole fraction of 2*pi (radians).
> y is a natrual number, counting the number of yellow circles.
> There are twice y number of x angles within the circle.

arctan(r) = (4*pi) / y
r = tan((4*pi) / y)
y = (4*pi) / arctan(r)

red_area = pi
yellow_count =
yellow_area = y * pi * r^2
blue_area_whole = pi * (1 + r)^2
blue_area = blue_area_whole            - red_area - yellow_area
          = (pi * (1 + r)^2)           - pi       - (y * pi * r^2)
          = pi * (1 + r + r^2)         - pi       - (y * pi * r^2)
          = pi + (pi * r) + (pi * r^2) - pi       - (y * pi * r^2)
          = pi + (pi * r) + (pi * r^2) - pi       - (((4*pi) / arctan(r)) * pi * r^2)

> The maximum occurs at a value for r where the derivative of
> blue_area with respect to r is zero.

blue_area'r (according to WolframAlpha):
         /     4 * pi * r^2             8 * pi * r    \
    = pi |--------------------- + 2*r - ---------- + 1|
         \(r^2 + 1)*arctan(r)^2          arctan(r)    /

> WolframAlpha says there are no real solutions where the derivative
> is zero, which suggests there is no value of r that maximizes the
> blue area.