Question 663002
A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles.
 Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow
:
A rough illustration of the end of a gutter; d=depth: d|__|d, call the bottom w
:
2d + w = 18
w = (18-2d)
:
Area:
A = d * w
replace w with (18-2d)
A(d) = d(18-2d)
A(d) = -2d^2 + 18d, the area as a function of the depth
:
The max area will be when d = the axis of symmetry, (x = -b/(2a)), 
therefore when a=-2, b=18
d = {{{(-18)/(2(-2))}}}
d = {{{(-18)/(-4)}}}
d = +4.5 inches is the depth which gives the max area (40.5 sq/in)
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Hey! did this make sense to you, any questions?