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Packaging Efficiency: Significance & Calculation of the Maximum Volume of a Rectangular Box

Packaging Efficiency: Significance & Calculation of the Maximum Volume of a Rectangular Box. Packaging Efficiency. What is it? An efficient, cost-effective way to package products Why is it important to us?

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Packaging Efficiency: Significance & Calculation of the Maximum Volume of a Rectangular Box

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  1. Packaging Efficiency: Significance & Calculation of the Maximum Volume of a Rectangular Box

  2. Packaging Efficiency • What is it? • An efficient, cost-effective way to package products • Why is it important to us? • Goods are individually wrapped or put into containers before they are sold to customers • Even before that, the goods are packaged in large numbers to be shipped to shops • With internet shopping, the goods must be delivered safely to the customers. • Packaging affects us frequently.

  3. Most Efficient Box - Overview • Mathematical Concepts Used • Volume of a Rectangular Prism • y = length * width * height • Polynomial Functions • Expanding/Factoring • Domain/Range • Derivatives • Power Rule • Critical Point (local extrema)

  4. Analyzing the Packaging Efficiency of an Existing Box 3-D Diagram of the Box

  5. Flattened Box with Measurements

  6. Rectangular Sheet of Box

  7. Calculations – Volume Function 2. 4. 1. 3.

  8. Domain & Range – Volume Function • Actual • Domain: • Range: • Unrealistic – negative/large values • Realistic • Domain: (0, 10.45) • Dependent on the limited surface area • Range: (0, 422.11] • Dependent on the domain

  9. Graph of Volume Function Different Width on the Volume of the Box volume (in cm³) width (in cm)

  10. Calculations - First Derivative of Volume Function →

  11. Calculations - Finding Ideal Dimensions (Max Volume) 3. 1. 2. 4.

  12. Ideal Dimensions (Max Volume) Different Width on the Volume of the Box volume (in cm³) width (in cm)

  13. Analysis – Percent Difference 2. 1. Actual Volume:

  14. Summary of Example • Actual Volume: 417.63cm³ • Maximum Volume: 422.11cm³ • Percent Error: 1.06% • Company did an excellent job in packaging its product efficiently.

  15. Other Forms of Packaging • Rectangular boxes aren’t the only shape that packages come in. • Most common though • Easy to stack neatly in trucks • Cylinders • A = πr²h • Cubes • A = s³

  16. Significance of Packaging Efficiency • Internet Shopping Malls • Goods need to be packaged • All goods are delivered • Efficient packaging reduces costs • Companies/Businesses • Goods need to be packaged • Efficient packaging reduces costs • Environmentally Friendly • Less materials used if packages become smaller • *Study published in 1995 indicated that this is one of the best ways to improve the environment.

  17. Works Cited Arnette, Sarah. "Business Definition of Packaging Efficiency." EHow. 13 Mar. 2010. Web. 6 Feb. 2011. <http://www.ehow.com/about_6068926_business-definition-packaging-efficiency.html>. "The First Derivative: Maxima and Minima." Calculus Tutorial - Harvey Mudd College Mathematics Department. Ed. Michael E. Moody. Harvey Mudd College. Web. 6 Feb. 2011. <http://www.math.hmc.edu/calculus/tutorials/extrema/>. The ULS Report. A Study of Packaging Efficiency As It Relates to Waste Prevention. Rep. Robert Lilienfeld and LARC Associates, Feb. 2007. Web. 7 Feb. 2011. <http://www.americanchemistry.com/plastics/doc.asp?CID=1593&DID=6072>. "Three Dimensional Box Applet." [MSTE] Office of Mathematics, Science, and Technology Education, University of Illinois. University of Illinois. Web. 6 Feb. 2011. <http://mste.illinois.edu/users/carvell/3dbox/box8by11.html>.

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