A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on e?

A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle 胃. How should 胃 be chosen so that the gutter will carry the maximum amount of water?A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on e?
It will fold like a trapezoid with bottom = 10 cm


Height = 10 sin 螛


Top of the trapezoid = 10 cos 螛 + 10 + 10 cos 螛


= 10 + 20 cos 螛





Area = (1/2)h(top + bottom)


A = (1/2)(10 sin 螛)(10 + 20 cos 螛 + 10)


A = 5 sin螛 (20 + 20 cos 螛)


A = 100 sin螛(1+cos螛)


A' = 100(cos螛)(1+cos螛) + 100(sin螛)(-sin螛)


0 = 100(cos螛+cos虏螛) - 100(sin虏螛)


100(sin虏螛) = 100(cos螛+cos虏螛)


1-cos虏螛 = cos螛+cos虏螛


0 = 2cos虏螛+cos螛-1


0 = (2cos螛 - 1)(cos螛 + 1)





cos螛 = 1/2 or -1


螛 = 蟺/3 or 蟺





The angle should be 蟺/3 because 蟺 will give zero area.


You can use second derivatives to prove 蟺/3 is the maximum point.