Question

A rain gutter is made from sheets of aluminum that are 18 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.
A. 5 inches
B. 4 inches
C. 4.5 inches
D. 5.5 inches

Answer 1

From the diagram you see that the cross-section is a rectangle
\[A = h*x\]
We want to find "h" that maximizes Area
solve for x in terms of h from total width equation:
\[x+2h = 18 \rightarrow x = 18-2h\]
Now we have Area in terms of h
\[A = h(18-2h) = 18h -2h^2\]
Set derivative equal to 0 to find maximum
\[\frac{dA}{dh} = 18-4h = 0\]
Finally solve for h.