a long strip of sheet metal 12 inches wide is to be formed into an open gutter with a rectangular cross-section as shown. what is the maximum possible area of the cross-section?

Answer 1

Bend it down the middle and form into a trough with angle Ø
The area of the cross-section of the trough will be

(1/2)(12)(12) sinØ
= 72 sinØ
d(area)/dØ = 72 cosØ
= 0 for a max of area
72 cosØ = 0
cosØ = 0
Ø = 90° or 270° , the last can be rejected

So when Ø = 90°
the area of the cross section will be
(1/2)(12)(12)sin90 = 72 square inches