OPTIMIZATION: Minimizing Packaging Costs

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SUMMARY

The discussion focuses on optimizing the cost of constructing a rectangular box with a square base and a fixed volume of 20 ft³. The cost per square foot for the base, sides, and top are $0.30, $0.10, and $0.20 respectively. Participants emphasize the need to derive a correct cost function based on the dimensions of the box, specifically using variables for the base length (x) and height (y). The correct formulation of the cost function is essential for applying optimization techniques to minimize packaging costs.

PREREQUISITES
  • Understanding of basic optimization techniques
  • Familiarity with cost functions and their derivation
  • Knowledge of volume calculations for rectangular prisms
  • Ability to differentiate functions for optimization
NEXT STEPS
  • Develop a cost function for a rectangular box with a square base
  • Learn how to apply the first derivative test for optimization
  • Explore the method of Lagrange multipliers for constrained optimization
  • Study real-world applications of optimization in packaging design
USEFUL FOR

Students in mathematics or engineering fields, particularly those focusing on optimization problems, as well as professionals involved in packaging design and cost analysis.

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Homework Statement


A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs $0.30/square foot, the material for the sides costs $0.10/square foot, and the material for the top costs $0.20/square foot, determine the dimensions of the box that can be constructed at minimum cost.


Homework Equations


Optimization.


The Attempt at a Solution


I know basic optimization... I will take the derivative to find a candidate for a relative minimum, and then verify it... But I REALLY need help trying to find the formula that I will use. I cannot seem to derive it from the information given. Any help is appreciated.
 
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Start out by labeling the variables that you need to determine the shape of the box. Then figure out formulas for the areas of the top and sides in terms of those variables. Then use that to write down a cost function to minimize. If you still have questions after you've shown us that much, I'm sure someone will help. You've got to at least START the problem.
 
My attempt...

The volume of the box would be found by x2y.

So the cost function should be...

C(x) = .30x * .10y * .20z

or

C(x) = .30x2 * .10y * .20z

Then I would optimize... But I am having trouble determining if I have derived the correct equation...

Can anyone help?
 
You don't have a correct cost function. Ok, so x is the length of the base and y is the height. I'm just guessing about that since you didn't say so. I have no idea what z is supposed to be. What are i) the area of the base and top and ii) what is the area of the sides? Multiply them by cost per unit area and add them up. Then you will have a correct cost function.
 

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