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.

 

[Architeuthis Dux]

I would recommend conducting a series of experiments to determine the optimal depth for the aluminum gutter. This could involve creating different gutter prototypes with varying depths and testing their water flow capacity under controlled conditions. Other factors such as the angle of the gutter, surface tension of water, and potential obstructions should also be taken into consideration. Additionally, mathematical calculations and simulations could be used to further analyze the relationship between depth and water flow. Overall, a combination of empirical data and theoretical analysis can provide a comprehensive understanding of the maximum water flow from a 12" aluminum gutter.