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Mathematics Governor’s Institute 2003

 

(Download as Microsoft Word documents: OPP, Worksheet, You Will Not Believe This)

 

Title of Project:  Optimal Product Packaging (OPP) using Volume & Surface Area

 

Team Members:  Tammy Baumann, James Blanchard, Michael Cutter, Bill Feeley, 

                             Rodney Hart

 

Grade Level and/or Courses:  9th/10th grade Geometry

 

Concepts Used:  Volume:  Surface Area of 3 Dimensional Shapes, Metric Measurement,

                                           Cost Analysis

 

Pa Standards Addressed:  2.2.11A; 2.3.8D; 2.3.11C; 2.4.11B & E; 2.5.11A, B, C, D;
                                          2.9.8D; 2.9.11A, I

 

NCTM Standards Addressed:  Geometry #1 & 4, Measurement  #2,  Data Analysis #3,
                                                  Problem Solving,  Connections

 

Introduction/ Application:  Volume and Surface Area of a cylinder is the topic of the classroom discussion.  We will explore as a class how to minimize surface area given a specific volume.  The students will already understand all of the volume and surface area formulas.  Students will already understand how to construct three dimensional shapes. 

 

Question:  What is the optimal shape, when the surface area is minimized and the
                  volume of 355 cm3 is maintained?

 

Model:  Various packaging formats for liquids.

 

Resources and Materials (estimated costs):  Worksheets (Surface Area and Volume, You will not believe this), Cardstock, Tape, Scissors,
                                                                        Rulers, Compasses

 

Procedures & Activities:  Short demonstration of calculations of minimum surface area
                                         with a given volume of 355 cc performed on a cylinder.  Class
                                         will go through all discussion and calculations for the cylinder.
                                          Students will be put into groups and given class time to
                                          work on the worksheet.

 

Answers: 

Cylinder;  Surface Area =  277.9 cm2 

                 Radius = 3.83 cm

                 Height = 7.72 cm

 

Rectangular Prism;        Surface Area = 300.76 cm2 

                                    Length = 7.08 cm

                                    Height = 7.08 cm

                                                            Width =  7.08 cm

                        Square Pyramid:           Surface Area =331.09 cm2

                                                            Length/ Width = 9.08 cm

                                                            Height = 12.92 cm

 

                        Sphere: Surface Area = 242.46 cm2

                                                Radius = 4.39 cm

 

                        Cone:               Surface Area = 305.48 cm2

                                                Radius = 4.94 cm

                                                Height = 13.88 cm

 

                             ** These answers are optimal and student answers may very.

 

 

Rubric:

                                    POINTS                      DESCRIPTION

 

                                         4                 Arrives at answers on worksheet with all work
                                                            shown.  Completes models of all geometric shapes.

 

                                         3                 Arrives at answers on worksheet (may or may not
                                                            show work). Attempts Geometric models.

 

                                         2                 Arrives at answers with no work shown on
                                                            worksheet.

 

     1                 Produces some solutions.  Worksheet not complete
                        and no models attempted.
    

0                                  Off task.  No work on worksheet and no models.



Accommodations/Adaptations: 

 

ESL:  Place students in small groups, paired with a student fluent in both languages

Special Ed:  Multiple choices for dimensions provided on the worksheet.
Enrichment:  Each group presents conclusions using technology.  Organize
                        data and chart results.

 

***Note***:

This is an exploratory exercise in which students are developing solutions.  They may not arrive at the “best possible” solutions that have been provided. It is hoped that all students will arrive at the conclusion that the sphere is the most cost effective.

 

 

 

***Extension***:

The next logical step would be to have students use the graphing calculator and arrive at the optimal dimensions that maintain the constant volume and minimize the surface area of each solid.

In a deeper extension, classes would attempt to derive equations which would provide all solutions; through graph analysis on their graphing instrument, they will extract the optimal answers.  The following equations provide the solutions for the cylinder, square pyramid , and cone:

CYLINDER: Y = (710/X) + 2PX2 ; where Y is Surface Area and X is the radius

 

SQUARE PYRAMID: Y =  X2 + 2X[(1134225/X4) + (1/4)X2]1/2 ; where Y is Surface Area and X is the length of one side of the square base

(This equation can be simplified; however, students are able to follow its derivation in this form.)

 

CONE: Y = Pr2 + Pr[r2 + (1065/Pr2)2]1/2 ; where Y is Surface Area and X is the radius 

(This equation can be simplified; however, students are able to follow its derivation in this form.)

 

 

Other applications would be presentations using PowerPoint or other available media.

 

 

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