Optimizing the Cross Section of a Gutter - Term Paper Example

Optimizing the Cross Section of a Gutter Introduction: Optimization is the philosophy of extracting maximum value from limited resources. One would always like to use less of material and form a component that provides maximum utility or functionality. This is natural because the resources are limited and the needs to be fulfilled are unlimited. Therefore, optimization is a must. Mathematics extends great help in the optimization process. This is done by finding out maxima of the output from a fixed input or conversely minima of the input for a desired but fixed output so that essentially the ratio of output to input is maximized. In this work the cross-section of a gutter has been optimized to contain maximum water with a fixed amount (length) of the material of construction. Problem Statement: A gutter is to be made of plastic to catch water at the edge of a roof. Naturally the manufactures would like to have a design of a gutter that hold maximum of water with a given amount of material required to fabricate the gutter. Now the problem is what should be the shape of the cross section of the gutter. Three possible shapes are presented below. The problem is to find out the optimum cross section which requires minimum of the construction material to hold a given volume of water. Solution: This appears to be a problem of maxima and minima. Looking at the suggested cross – sections the problem reduces to minimizing the periphery of the gutter cross section for a given area of the cross-section, this is because, while the periphery is made of the construction material (here plastic) the cross-section hold the water. Analytical Solution Let us assume that the length of material (here plastic) is ‘l’ and this is folded to make different cross – sections for the gutter with area ‘A’. Now for a given area ‘A’, the length of the material ‘L’ should be minimized or for a given length of the material ‘L’, the cross-sectional area ‘A’ should be maximum. In this analysis, four shapes will be considered – Triangular cross - section, Rectangular Cross –section, Semicircular cross – section and circular cross – section. Triangular cross-section: Let us assume that the material of length ‘l’ is folded to make a triangular cross-section with area ‘A’ and included angle ‘’. Therefore, A = (1/2)*(L/2)*(L/2)*Sin For A to be maximum for a given ‘L’, there is only one possibility of varying . For A to be maximum or minimum i.e. Sin = 1. Therefore, the maximum possible value of area ‘A’ will be A = (1/8)L2 For Rectangular cross-section Let us assume that height of the rectangular gutter is ‘x’, then length of the gutter will be (L-2x)/2 and hence the area will be A = x*(L-2x)/2 =(1/2) Lx – x2 For, A to be maximum or minimum Therefore, maximum area of the rectangular gutter will be, A = (1/16)L2 Semi-circular cross-section Radius will be r = L/ and Areas will be A = (1/2)r2 = (1/2)L2 = (1/6.28)L2 Circular cross-section Radius will be r = l/() and Areas will be A = r2 = (1/4)l2 = (1/12.56)L 2 These values are presented in the form of a table in Table 1 for the sake of comparison. Table 1: Area of cross-section of gutter with a given periphery ‘L’ Shape of the Gutter Area of Cross-section Triangular (1/8)L2 Rectangular (1/16)L2 Semicircular (1/6.28)L2 Circular (1/12.56)L2 Conclusions: Based on the analysis of the problem it can be concluded that the gutter should have semi-circular cross-section to minimizes the use of construction material. Read More