Calculus I Maple Project on Packaging: For a Fixed Volume, Finding the Minimal Surface Area That Contains the Volume
Packaging is a huge business worldwide. Concerns of marketing and economics require precise calculation of package dimensions in order to be “competitive.” Suppose that we want to construct a package that will hold a fixed volume; how can we do this most efficiently so that the amount of material used, measured as surface area, is as small as possible. In this project, we will study this problem for two types of packages, the cylinder and the rectangular box with a square base.
Part I: Working with a right circular cylinder
The table below gives the approximate measurements in inches of several common items packed in cylindrical containers.
Product Radius (in.) Height (in.) Volume (in.3) Baking Powder 1.25 3.65 17.92 Cleanser 1.45 7.50 49.54 Coffee 1.95 5.20 62.12 Coffee Creamer 1.50 6.85 48.42 Cake Frosting 1.63 3.60 30.05 Orange Juice 2.10 6.70 92.82 Soup 1.30 3.80 20.18 Tomato Puree 1.95 4.40 52.56
1. The equation for the surface area of a right circular cylinder is easy to obtain. The enclosing area consists of the areas of two circles top and bottom and the area of the “side,” which is a rectangle if you imagine slicing a label on a can vertically and then unrolling it. The dimensions of the rectangle are height h by circumference of the circle 2r. Adding this together, we obtain the equation
S=2r2 +2rh
Use the numerical data above to calculate surface areas for the containers listed, and fill in the table below. Round your calculations to the nearest hundredth.
Product Surface Area (in.2) Baking Powder Cleanser Coffee Coffee Creamer Cake Frosting Orange Juice Soup Tomato Puree 2. The baking powder container described in the data for this project has a radius of 1.25 inches and a surface area of 38.48 square inches, and a volume of 17.92 cubic inches. Is it possible to design another cylindrical baking powder container with the same surface area and volume but with a different radius? Explain why or why not. If it is possible, find the dimensions of another cylindrical container having a surface area of 38.48 square inches and a volume of 17.92 cubic inches.
3. Do you think that each of the products listed (cleanser, coffee, etc.) uses a container that minimizes surface area for the fixed volume given in the data? Why or why not?
4. Suppose you are designing a baking powder container that has a volume of 17.92 cubic inches. Use the equations for the surface area of a cylinder and the volume of a cylinder to obtain (algebra!!) an equation for surface area S in terms of radius r. The equation (formula) for the volume of a (right) circular cylinder is
V=r2h
Now use Maple to plot the graph of surface area as a function of radius, and use the graph to determine the value of the radius r that minimizes the surface area S. How does this minimizing radius compare to the radius given in the data? Does the baking powder container given in the project data minimize surface area?
5. Repeat Exercise 4 for each of the products below and record the “optimal” radius for making surface area as small as possible for a given fixed volume. Then use the volume and radius to determine the height and then the surface area for each container. Fill in the table and be sure to save the various graphs of surface area versus radius—they will become part of the Maple “report” that will be uploaded to your ePortfolio.
Product Volume (in.3) Radius (in.) Height (in.) Surface Area (in.2) Baking Powder 17.92 Cleanser 49.54 Coffee 62.12 Coffee Creamer 48.42 Cake Frosting 30.05 Orange Juice 92.82 Soup 20.18 Tomato Puree 52.56 Part II: Working With a Rectangular Box
6. Design a rectangular container with a square base for each of the volumes and heights given below. The volume of a rectangular container with a square base of side length x and height h is given by
V=x2h and the surface area is given by
S=2x2 + 4xh
Compare the corresponding formulas for volumes and surface areas for cylinders and rectangular boxes, and note the similarities and differences. Fill in the table below.
Volume (in.3) Height (in.) Side of base (in.) Surface Area (in.2) 17.92 3.65 49.54 7.50 62.12 5.20 48.42 6.85 30.05 3.60 92.82 6.70 20.18 3.80 52.56 4.40
7. Imitating your method in Part I Exercise 4, manipulate the two equations above to find a formula for surface area S of a rectangular box in terms of the length of the side of the square base x. Use a similar graphical method to determine the optimal side length and the minimizing surface area for the fixed volumes above. Record your results below and save your Maple graphs as part of your ePortfolio project.
Volume (in.3) Optimal Side Length (in.) Minimized Surface Area (in.2) 17.92 49.54 62.12 48.42 30.05 92.82 20.18 52.56
8. Compare the surface areas from the tables in Exercise 6 and Exercise 7. Do consumer products “on the market” always have optimal smallest surface area? Marketing question: why might a company choose not to go with least surface area for a given product volume? Try to give a couple of reasons.