Question

A rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

Answer 1

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:

$2D+W = 12\\W=12-2D\\A =D*W\\A(D)=-2D^2+12D$

The depth for which the derivate of the area function is zero is the depth that yields the maximum cross-sectional area:

$A'(D)=0=-4D+12\\D=3\ inches$

The cross-sectional area for D = 3 is:

$A(3)=-2*3^2+12*3\\A(3) = 18\ in^2$