A rain 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