Surface Area and Volume Applications Worksheet

 

Acme Packaging Company produces 12 ounce aluminum cans for Pepsi, Coke, and all other major bottlers.  Cost analysis engineers have recently become concerned about the increasing cost of making the cylindrical aluminum cans.  They have decided to investigate the cost of producing alternative containers for the 12 ounce soda products.

 

Below is a lab in which you are asked to calculate and make models for several alternative possibilities to contain the soda.  Show all work as you work through the lab.

 

1.  Calculate the surface area of a typical can ( you can assume that it is a perfect cylinder)                                                                                                             6.2 cm

 


           

                                                                                                        11.76 cm

 

 


 Surface Area =                                              

 

 

 

NOW CONSIDER THE FOLLOWING SHAPES:

Square Pyramid, Cone, Sphere, Rectangular prism.

 

 For each shape you must find the dimensions that would allow the container to hold the 12 ounces(355cc) of liquid, but minimize the amount of aluminum to build the container (minimize the surface area).  Do all of your work for each shape on a single sheet of lined paper.  One piece of paper will be used for each shape with the name of the shape at the top of the paper.  Staple all four of the sheets to the back of this worksheet.  Then below write your final answers for each shape in the spaces provided. 

 

REMEMBER, YOU MUST SHOW ALL WORK FOR EACH SHAPE ON A SEPARATE SHEET OF PAPER WITH A TITLE!!

 

 

2. Rectangular Prism: Select a length, width and height,

but keep the volume the same:

 

Length = ______   Width = ______  Height =  ______

 

Volume = l*w*h =355

 

Surface Area= 2lh + 2wh + 2lw = _________

 

 

                       

 

  1. Square Pyramid: Now select a length and height, and again, keep the volume equal to 355 cc.

                                                                                                            h         

Length = _____ Height = _____ Slant Height = _____

 

Volume = 1/3  P r2 h = 355                                                                                       s

 

Surface Area = 4( ½ l s) + s2  = _______                                                         l

                                                                                                                 

                                                                                                                                     

4.  Cone: Select a radius and height, keeping the volume equal to 355 cc.                                                                    

 

Radius = _____ Height = ______ Slant Height = ______

 

Volume = 1/3 P r2 h = 355                                                                s               h

 


Surface Area = P r2 + P s r = ________                                                                          r

                                                                                                                                     

 

 

5.  Sphere: Find the radius that will give a volume of 355cc.

 


Radius = _______

                                                                                                                                         r

Volume = 4/3 P r3  = 355                                                                                 

 

Surface Area = 4P r2 = _________

 

 

6.  Consider all five shapes that will hold 12 ounces (355 ml).  Which shape is the most cost effective?  __________________________  Why?  _________________________
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7.  Assume Coca Cola ships  3 billion cases( 24 cans per case) worldwide last year.  Using the most optimal shape, what would Coke save per year? ________________________

SHOW WORK BELOW

 Hint:  calculate the total amount of surface area for a case of cylindrical cans and compare it to the total surface area of the optimal containers