Gutter Design - Math Project

Dear Welbilt Design and Construction,

Eby Enterprises is happy to provide the following analysis for your review. Our mathematicians worked tirelessly to assist you within your time constraints and provide this detailed comparison, reference, and recommendation. You will find within our report that a gutter with equal length sides and equal bends of sixty degrees up will provide the closest to optimal area.

We look forward to continued business with Welbitl Design and Construction. Our calculations are always at your disposal, no matter the task or time constraint.

Sincerly,

The mathematicians of Eby Enterprises

Contents

1. Graphics of gutters

2. Rectangular gutter analysis

3. Differing side length trapezoidal gutter analysis

4. Equal length side trapezoidal gutter analysis

5. Varriance from optimal and conclusion

Graphics and Constraints

W=35.7

Area=202.853

b=9.566

W=15.088

Area=170.748

b=8.925

W=17.85

Area=159.311

b=10.306

W=11.9

Area=183.957

The Area of the gutters cross-sectional is limited by the 35.7 inches of metal and the three sides, created with two bends.

Rectangular Gutter Analysis

Our first analysis uses two simple bends of 90 degree angles to form a rectangular gutter with sides of b height and a base of W width. In this scenario, the width W of the gutter will always be 35.7 minus twice the height b, W=35.7-2b. The area for this rectangular gutter is simply a product of height and width, AreaOfRectangleb=35.7b-2b2. To find the maximum area of this rectangular gutter, we use previously shown equation to find the following derivative and solve for zero, AreaOfRectangle'b=35.7-4b.

35.7-4b=0

b=-35.7-4

b=8.925

Solving the area of the rectangular gutter with the maxima found from the derivative AreaOfRectangle8.925=35.78.925-28.9252, we find the optimal rectangular gutter will measure 8.925 inches in height and 17.85 inches in length (35.7-w8.925=17.8) and have an area of 159.311 square...