Arain gutter is made from sheets of aluminum that are 20 inches wide by turning up the edges to form right angles. determine the depth of the gutter that will maximize its cross sectional area to allow the greatest amount of water to flow

For the answer to this question,

Let x = depth of the gutter
then
18 - 2x = width of the gutter
Cross section area 
A = depth  x  width

 = x(18 - 2x)
A = 18x - 2x^2
:
Axis of symmetry will give x value for max area:

x = -b/(2a)
x = -18/(2x(-2))

x = 4.5 inches is the depth for max area

Max area: 4.5 * 9 = 40.5 sq/in

I hope my answer helped you.

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