rainflow is a new U-shaped system of guttering. The cross-section is a semi-circle with radius x topped by an open rectangle of heigh h. The total distance ABC is 20cm. For which values of x and h will Rainflow carry the most water.
You don't say what ABC is. I'll assume it is the perimeter of the cross-section.
So you want the maximum area when 2h+πx = 20
The area is, of course,
A = 2xh + 1/2 πx^2
= 2x(20-πx)/2 + 1/2 πx^2
= 20x - π/2 x^2
dA/dx = 20 - πx
so the maximum area of 200/π occurs when x = 20/π
That makes h = (20-π(20/π))/2 = 0
so the maximum water volume occurs when the gutter is just a semi-circle.
I guess I didn't pick the proper meaning for ABC...