Arain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90 degrees. what depth will provide maximum cross-sectional are and hence allow more water to flow? what depths will allow at least 16 square inches of water to flow?

2x + y = 12 
y = 12 - 2x 

A = x * y 
A = x * (12 - 2x) 
A = 12x - 2x^2 

Derive 

dA/dx = 12 - 4x 

Set dA/dx = 0 

0 = 12 - 4x 
4x = 12 
x = 3 

A = x * y 
A = 12x - 2x^2 
A = 12 * 3 - 2 * 3^2 
A = 36 - 2 * 9 
A = 36 - 18 
A = 18 

So 18 square inches is the largest cross-section 
x = 3 , y = 6 
Those are your dimensions .

16 = x * y 
16 = 12x - 2x^2 
2x^2 - 12x + 16 = 0 
x^2 - 6x + 8 = 0 
x^2 - 6x + 9 = 1 
(x - 3)^2 = 1 
x - 3 = -1 , 1 
x = 2 , 4 

2 inches or 4 inches will give you a cross-section with an area of 16 
2 </= x </= 4 is the domain