Answer:
Depth = 3 inches
Maximum cross-sectional area = 18 inches²
Step-by-step explanation:
Let 'D' be the depth of the gutter and 'W' be the width of the gutter, the cross-sectional area as function of depth, A(D), is:
[tex]2D+W = 12\\W=12-2D\\A =D*W\\A(D)=-2D^2+12D[/tex]
The depth for which the derivate of the area function is zero is the depth that yields the maximum cross-sectional area:
[tex]A'(D)=0=-4D+12\\D=3\ inches[/tex]
The cross-sectional area for D = 3 is:
[tex]A(3)=-2*3^2+12*3\\A(3) = 18\ in^2[/tex]
