Expert: Scotto - 10/2/2008

Question
Dear EB Enterprises: Welbilt Design and Construction has recently begun the construction of a series of carefully designed, eminently safe new office buildings for Duff’s. At the last minute, however, the subcontractor we had working on the design of the gutters for the roofs of building has pulled out, so we are contacting your firm to determine the optimum gutter design for the development. An optimum gutter will, of course, carry a maximum amount of water, and our design constraints require that it be manufactured from a 19 inch wide piece of material (which we can order it in the lengths required to line different roof sections). Due to a machine limitation, we MUST make two folds, or bends in the width of the metal to form the shape of the gutter. We can make those folds at anyangle needed, however. We therefore have contacted you to find the gutter design(s) which will carry the most water from the roof. Because we have already begun the construction of the office buildings, we unfortunately require a fairly speedy response from you, and therefore MUST have your report by the deadline. We are not able to give you the perimeter of the roofline, since the final design keeps changing. (Those darn stonecutters) So is this job even possible? If it is, it would really help if you could give us at least two different options. It would also be nice to know how far off from optimal we are.
For your reference, all reports submitted to Welbilt, Inc. should be written so that the forewomen and foremen of the construction unit implementing the report can understand and apply the information contained therein. Owing toWelbilt’s preeminent position in the construction field all of our forepeople have degrees in engineering, and thus have had college level mathematics, including calculus—unfortunately, however, their long experience in the field (and extensive time at Moe’s) precludes a ready knowledge of the same. Therefore, the reports should assume a strong precalculus background, but should notexpect a knowledge of much more than that. We look forward to seeing your finished report.


Answer
If the material is 19" wide, we must put x inches in the base and (19-x)/2 in each side.  I believe the optimal result is to make square corners.

The area of the cross section of a cyllinder is x(19-x)/2.  To maximize it, set the derivative equal to 0.  That is, find the derivative of (19x-x²)/2 and set it equal to 0.  The derviative is (19-2x)/2.  Since we are setting it equal to 0, the 1/2 can be dropped, giving 19-2x=0.  Solve for x, find the height, and that's the answer.