Question

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

greatest carrying capacity ---> greatest volume.
Since the volume = 120(area of end cross-section)
the greatest volume is a function of the surface area of the end
let the length to be folded up be 4 inches, with an angle of Ø

looking at one of the two equal right-angled triangles,
let the height be h
sinØ = h/4
h = 4sinØ -------> we need that depth !
base of triangle -- b
cosØ = b/4
b = 4cosØ

SA = 2 triangles + rectangel
= 2(1/2)(4cosØ)(4sinØ) + 4sinØ(4)
= 16sinØcosØ+ 16sinØ
= 16sinØ(cosØ + 1)

d(SA)/dØ = 16[ sinØ(-sinØ) + cosØ(cosØ+1)]
= 0
-sin^2Ø +cos^2 Ø + cosØ = 0
cos^2Ø - 1 + cos^2Ø + cosØ = 0
2cos^2Ø + cosØ - 1 = 0
(2cosØ - 1)(cosØ + 1) = 0
cosØ = 1/2 or cosØ = -1, the last one would be a fold of 180°, ---> not very likely

if cosØ = 1/2, Ø = 60°
and h = 4sin60°
= 4(√3/2)
= 2√3 inches or appr 3.464 inches