Background:
Cans are amazing. Cans come in many sizes and are typically made of either steel or aluminum. Explore this website www.cancentral.com to learn more about cans.
Cans are a component of the cost of providing food. Can manufacturers (and ultimately the companies that put things INTO the cans) should want to minimize the amount of steel or aluminum in a given volume of can to hold down the cost of the can. (Assume that the volume of any particular can is constant. For example, the Campbell's Soup Company wanted a can that would hold exactly 318 ml of soup.) The amount of material used in the can is equal to the surface area of that can. Finding the minimum of the surface area of a can for a given volume is an optimization problem that uses calculus.
Our goal in this project to analyze a given can to see if it is optimal (in terms of surface area) and if not, to see how far off optimal it may be.
Your Tasks:
1) CHOOSE a canned product (food or otherwise). Find the volume and surface of your actual can.
2) WRITE a formula for the surface area of a can with that volume. Your surface area function will originally have two variables (height and radius). Remember, we are holding the volume constant. So use that fact to write the height in terms of the radius and then use that to eliminate the height as a variable in the surface area function.
3) GRAPH the surface area function.
4) Do an "OPTIMIZATION ANALYSIS" :
a. find the radius that optimizes the surface area of the can. Your real world constraint here is that your radius cannot be less than zero. (It's difficult to have a can with negative radius.) .
b. find the surface area of your optimal can and compare that to the surface area of your actual can.
5) Do a "FIRST DERIVIATIVE ANALYSIS" on the Surface Area Function. Recall that a "First Derivative Analysis" is where you find all intervals where the surface area function is increasing or decreasing and where you find all relative extrema. (NOTE 1: For this First Derivative Analysis, treat the SA function as a function where r can be less than zero.) (NOTE 2: Remember to not just show the first derivative chart, but to also write English sentences explaining your response.)
6) Do a "SECOND DERIVATIVE ANALYSIS" of the Surface Area Function. Recall that a "Second Derivative Analysis" is where you find all intervals where the surface area function is concave up or concave down and where you find all points of inflection. (NOTE 1: As above, for the purposes of this Second Derivative Analysis, treat the SA function as a function where r can be less than zero.) (NOTE 2: Again, remember to show the 2nd derivative chart and to write English sentences explaining your response.)
7) BUILD a paper version of the optimum can, and line it up next to your actual can. Take a photo of the two cans together or otherwise create an actual display of the two cans side-by-side.
8) ANALYZE why the actual can is designed as it is. If the can is not optimal, try to explain why the can is sub-optimal. If the can is optimal in terms of surface area, determine whether there are other factors that may improve the design of the can from perspectives other than optimizing surface area. Write your response with well thought out paragraphs and full sentences.
Compile all your work together neatly in a written form (use the eight parts above to help your organize your final product) .