Let the radius of the circle be r and each side of the square be x
so 2πr + 4x = 12
πr + 2x = 6
x = (6-πr)/2
Area = πr^2 + x^2
= πr^2 + ((6-πr)/2)^2
= πr^2 + (3 - πr/2)^2
d(Area)/dr = 2πr + 2(3-πr/2) (-π/2)
= 2πr - π(3-πr/2)
= 0 for a minimus Area
2πr = π(3-πr/2)
2r = 3-πr/2
4r = 6 - πr
4r+πr = 6
r = 6/(4+π) = appr .840
so we need 2πr or 5.2788 ft for the circle,
leaving 6.721 ft for the square.
radius of circle = .840 ft
side of each square = 1.68 ft
2πr + 4x
= 2π(.840) + 4(1.68) = 11.998 , not bad
let r = .8 , then x = 1.743
area = π(.8)^2 + 1.743^2 = 5.049
let r = .9, then x = 1.586
area = π(.9)^2 + (1.586)^2 = 5.06.
r = .84 , x = 1.68
area = π(.84)^ + 1.68^2 = 5.039 , which is lower than either of the
slightly larger and slightly smaller radii.
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