A 10-foot section of gutter is made from a 12-inch-wide strip of sheet metal by folding up 4-inch strips on each side so that they make the same angle with the bottom of the gutter. Determine the depth of the gutter that has the greatest carrying capacity.

Answer 1

If the angle is x, then you have a trapezoid with lower base 4", and upper base 4+2*4cosx, and height 4sinx.

So, the area of the cross-section is

a = (4 + (4+8cosx))/2 * (4sinx)
= 16 sinx (1+cosx)

da/dx = 16(2cos^2(x)+cos(x)-1)
= 16(2cosx-1)(cosx+1)

da/dx=0 when cosx = 1/2, or x = 60°

Check to be sure it is a max, not a min.