Coca-Cola is one of the biggest companies in the world. It’s also one of the most successful, raking in billions of dollars every year. But is it overlooking a cost-saving opportunity in something as basic as the design of its classic can?

In this lesson, students will use surface area and volume to model the cost of a soda can. Then, they’ll come up with a rational function to search for a can design that’s even less expensive to make than the current one.

### Students will

• Calculate surface area and volume for different cylinders
• For a fixed volume, explore how surface area varies with the shape of a can and relate this to cost
• For a fixed volume, build a function for the surface area of a cylinder in terms of its radius
• Use technology to find the radius that minimizes surface area of a cylinder for a given volume
• Discuss why soda cans might be shaped the way they are

### Before you begin

Students should know the formulas for the volume and surface area of a cylinder (or be able to derive them). It’s also important for them to know how to solve for one variable in an equation in terms of another — for example, given the formula for the volume of a cylinder, they should be able to solve for the height of the cylinder in terms of the volume and the radius. This lesson also serves as a nice application of rational functions, so it helps if students have seen these types of functions before and are able to graph them using technology.