The Guttering Problem

The problem is to design Rain Channel Guttering having a fixed cross-cectional perimeter, a straight base and two equal-lengthed straight sides such that it's Cross-Sectional-Area (CSA) is as large as it can be.  In this case it will allow the most rainfall to flow through it, whatever its physical length.

Drag

Drag

Hint

Locus

θ = 0.8726646259971648 rad (slider from 0 to γ, step 1*°)
P = 10 (slider from 0 to 10, step 0.1)
A = Intersect(xAxis, yAxis)
I = Point(Circle[A, P])
PL = Segment((x(A) + 0.01, 0), (x(I) - 0.01, 0))
B = Point(PL)
m = Distance(A, B)
n: (P - m) / 2
c = Circle(B, n)
C, D = Intersect(c, xAxis)
E = Midpoint(A, B)
B' = Rotate(B, 90*°, E)
b = Ray(E, B')
g = Circle(B, n)
F = Intersect(g, b)
yG = If(x(C) ≥ m / 2, 0, y(F))
xG = If(x(C) ≥ m / 2, x(C), m / 2)
G: (xG, yG)
k = CircleArc(B, D, G)
d = Circle(A, n)
f = Segment(B, A)
a = OrthogonalLine(E, xAxis)
e = Circle(E, 2 * n)
h = OrthogonalLine(D, xAxis)
α = Angle(B, E, B')
H = Rotate(D, θ, B)
H' = Mirror(H, b)
i = Segment(A, H')
j = Segment(B, H)
β = Angle(I, B, H)
GutterArea, h', h_1, b_1, a_1 = Polygon(H', H, B, A)
p = Circle(B, 1.5)
J = Intersect(PL, xAxis)
K = Intersect(PL, xAxis)
L, M = Intersect(p, xAxis)
l = Segment(B, M)
yPα: GutterArea
xP: m
xPθ: θ
CSA: (m, GutterArea)
loc1 = Locus(CSA, B)
γ = Angle(D, B, G)
AreaG(x) = (P - x) / 2 * (x + (P - x) / 2 * cos(θ)) * sin(θ)
q = Function(AreaG, 0, PL)
Root1 = Root(q, 0)
GRoot = Max(0, x(Root1))
Loc = Function(AreaG, GRoot, PL)
Base: m
Side: n
BtoS: Base / Side
text1: "Base : Side = " + Base + " : " + Side + " = " + BtoS + " : 1"
o = false
r = Segment(CSA, (x(CSA), 0))
s = false