Design of a Grain Hopper

by Jason Salm

Submitted : Fall 2011

This project is a great example of the kind of work an industrial engineer does. The way to solve this problem is to use the strategy of LaGrange Multipliers. I first used the diagram of the object to come up with the equations for the surface area (to minimize) and volume (the constraint). I then had to take the partial derivatives with respect to x, y, and z of both of those equations. Then I took the two partial derivatives of x and set them equal to each other and multiplied one side by lambda, and did the same with both y and z. After that I needed to solve those 4 equations (including my constraint) for x (dimension A), y (dimension B), z (dimension L), and lambda.

Using this process I found that the minimal dimensions of the diagram from the problem statement were: A=2 cube root(5), B=2 square root(2) cube root(5), z= 2 cube root(5), and lambda = 2/cube root(5). With these dimensions the minimal surface area would be 35.09m². Knowing what the minimum amount of material needed to produce this product and the dimensions needed to get this surface area, reduces cost for companies by a lot due to the reduced amount of materials needed.

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