I might bring in a soup can today to start a conversation.
"How much do you think it costs to make this can?"
"About how many cans of soup do you think are sold each day in the United States?" (The Campbell's Soup company alone claims to sell almost 2 billion cans of soup each year.)
"Suppose I could find a way to save a penny or two on each can. Would it be worth my trouble?" (If Campbell's could save a penny on each can, they'd save almost 20 million dollars in a year!)
Hand out the Tin Can problem.
Beginning individually or in small groups, students will need to make sense of the situation in terms of both cost and geometry. Beginning with a "verbal equation" like
Total Cost = 2(top of can)(price) + (sides of can)(price),
they should then progress to a full-blown rational function expressions the cost of a tin can in terms of its radius. This won't be easy, but be careful not to give it away! Creating the equation is the main focus of this problem, so you need to make sure that students are supported in their thinking, but that they're also completely responsible for making sense of the problem and persevering in solving it. (MP 1)
Once students have produced the correct function and a reasonable graph, they need to interpret the graph to identify the ideal can.
Bear in mind that finding the minimum cost analytically is beyond the scope of an Algebra 2 class. Students should make use of whatever tools are appropriate (MP 5) in order to estimate the actual minimum cost. This may include using graphing calculators, an online tool like GeoGebra, or simply evaluating the function at various r-values. In any case, the most important thing is that students are able to correctly interpret the lowest point on the graph in terms of the radius of the can and its cost.
As students wrap up this problem, an excellent extension is to ask how the situation might change if material costs increase. Can the company afford to keep changing the dimensions of the can to respond to fluctuating costs of raw materials? Can the company change the volume of the can to save money? You might use this extension problem for your advanced students.