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Carpe Donut in Charlottesville, Virginia, has an interesting pricing scheme. You can buy one donut for \$2, two donuts for \$3, three for…well, you get the idea. This means that two people could pay less by purchasing their donuts together. Three people could do even better. So how does the average cost per donut change, and how much should we be paying for each?

In this lesson, students use linear, rational, and piecewise functions to describe the total and average cost of an order at Carpe Donut. Through this, they’ll see the benefits of buying in bulk, and will find a least-expensive way to purchase these delicious donuts.

### Students will

• Use linear and piecewise-linear functions to describe a pricing scheme
• Model average cost using rational functions
• Make formal or informal limit arguments to explain end behavior of total and average cost functions

### Before you begin

Students should be able to graph and write equations for linear functions. They should also know how to calculate an average. Some previous exposure to piecewise-defined functions (the floor function in particular) will also be helpful. Limits make an appearance, but prior experience with them is not necessary. Lastly, although rational functions play an important role in the lesson, we assume no previous experience.