Max Water Flow from 12" Alum Gutter

[thesandsam]

i don't know if one could do this without the diagram but...
a rain gutter is to be made of aluminum sheets that r 12 inches wide by turning the edges 90 degrees. what depth would provide maximum cross-sectional area and hence allow the most water to flow?

 

[mathmike]

lets see what we know

x = bottom of gutter

y = sides

so x + 2y = 12

a = xy

:y = 6 - x / 2

sub in equation for a

a = x ( 6 - (x/2))

a = 6x - (x^2 / 2)

da/dx = 6 - x

set this to 0 so that

6 - x = 0

x = 6

so the max will be at either the endpoints or at x = 6

so we have

6x - (x^2 / 2) = a

f(0) = 0 -0 = 0 no.

f(12) = 6(12) - (12^2 / 2) = 72 -72 = 0 no.

f(6) = 6(6) - (6^2 / 2) = 18 yes.