optimization problem
A very real example of max / mins is to determine the minimum amount of material required to produce an aluminum can. Considering the amount of soda consumed every year and the cost of refining aluminum, a very small precent error in the design of container can result in the loss of millions of dollars in materials and energy to industry each year, not to mention the environmental considerations. Now consider some of the constraints involved in soda can design. First the can must hold 354 cubic centimeters. Additionally, we must be able to set the can on a surface without it tipping over. Hence, it must have a flat bottom. The question of cross - sectional shape is another optimization problem itself, but assuming that a circle is the most efficient cross section (not to mention the most comfortable to hold in your hand), we will restrict our design consideration to right, circular cylinders. We begin by setting up the problem. Answer the following questions that might influence the design of a can. Given that we want the soda can to hold 354 cubic cm, what volume should the can have ? Given the right circular cylinder shape, what is the formula for the volume of the can ? What dimensions may we vary while still satisfying the volume and shape requirements ? Can we express one dimension (variable) in terms of the other ? What are we trying to minimize when we manufacture soda cans ? What is a mathematical expression for the answer to question 3 ? How many variables docs it have? Can we write it in terms of one variable ? How can we optimize (minimize) the function in question 4 ? Compare your solution for the soda can design with the true dimensions of a soda can. How do they differ ? Is the beverage industry woefully negligent in wasting aluminum in the construction of cans ? Why aren't environmentalists, who are. no doubt, concerned with bauxite mining, up in arms about this waste of material ? Inquiring minds want to know ... What assumptions did you make that are inconnect? (Hint: Is the thickness of a soda can uniform ? ) What parts of a soda can arc thicker ?. Why ? Now that you've made it this far. I'll tell you: In order to accommodate the pull - ring, the thickness of the top of a soda can is three times thicker than the side and bottom of the can. Why ? Armed with this new fact, re - do the soda can problem, and determine the optimal design of the can. How do your new results compare to the actual dimensions of the can ? Make a close inspection of a soda can. What do you notice about the side of the can near the top ? Is the radius of the top equal to the radius of the majority of the can ? Why ? What do you notice about the bottom of the can ? The rounded indentation is there for a reason. What is the reason ?
