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.

What's an acceptable dating range? Students use linear equations and linear inequalities to examine the May-December romance, and ask whether the Half Plus Seven rule of thumb is a good one.

Topic:
Building Functions (BF), Creating Equations (CED), Expressions and Equations (EE), Functions (F), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

How much should people pay for donuts? Students use linear, rational, and piecewise functions to describe the total and average costs of an order at Carpe Donut.

Topic:
Building Functions (BF), Interpreting Functions (IF)

How do cell phone towers identify your location? Students describe geometrically the location information provided by a cell phone tower, explain why loci from at least three towers are required to pinpoint a customer's location, and consider the tradeoff between coverage and "locatability" when a phone company chooses a new tower location.

Topic:
Congruence (CO), Modeling with Geometry (MG)

How far away from the TV should you sit? Students use right triangle trigonometry and a rational function to explore the percent of your visual field that is occupied by the area of a television.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI), Similarity, Right Triangles, and Trigonometry (SRT)

How much do you really pay when you use a credit card? Students develop an exponential growth model to determine how much an item really ends up costing when purchased on credit.

Topic:
Building Functions (BF), Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE)

Could Inspector Javert have survived the fall? Students use quadratic models to determine how high the bridge was in Les Misérables, and explore the maximum height from which someone can safely jump.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF)

How much should you bid in an auction? Students use probability, expected value, and polynomial functions to develop a profit-maximizing bidding strategy.

Topic:
Building Functions (BF), Interpreting Functions (IF)

How has the iPod depreciated over time? Students compare linear and exponential decay, as well as explore how various products have depreciated and what might account for those differences.

What’s the best strategy for creating a March Madness bracket? Students use probability to discover that it’s basically impossible to correctly predict every game in the tournament. Nevertheless, that doesn’t stop people from trying.

Topic:
Conditional Probability and the Rules of Probability (CP), Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE)

In which Major League Baseball stadium is it hardest to hit a home run? Students find the roots and maxima of quadratic functions to model the trajectory of a potential home-run ball.

Topic:
Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

Does the same sound always sound the same? Students come up with equations in several variables to explore the Doppler Effect, which explains how sound from a moving object gets distorted.

How has the urban population changed over time, and will we all eventually live in cities? Students use recursive rules along with linear and exponential models to explore how America's urban areas have been growing over the last 200 years.

Topic:
Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE), Reasoning with Equations and Inequalities (REI)

How much should Nintendo charge for the Wii U? Students use linear functions to explore demand for the Wii U console and Nintendo's per-unit profit from each sale. They use those functions to create a quadratic model for Nintendo's total profit and determine the profit-maximizing price for the console.

Topic:
Creating Equations (CED), Building Functions (BF), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

Which size pizza should you order? Students apply the area of a circle formula to write linear and quadratic formulas that measure how much of a pizza is actually pizza, and how much is crust.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

How have temperatures changed around the world? Students use trigonometric functions to model annual temperature changes at different locations around the globe and explore how the climate has changed in various cities over time.

Topic:
Building Functions (BF), Interpreting Functions (IF)

How do vehicles turn? In this lesson, students use the geometry of circles to understand how we get from point A to point B when the path isn’t a straight one.

How should pharmaceutical companies decide what to develop? In this lesson, students use linear and quadratic functions to explore how much pharmaceutical companies expect to make from different drugs, and discuss ways to incentivize companies to develop medications that are more valuable to society.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

Do social networks like Facebook make us more connected? Students create a quadratic function to model the number of possible connections as a network grows, and consider the consequences of relying on Facebook for news and information.

Topic:
Building Functions (BF), Creating Equations (CED)

Why hasn't everyone already died of a contagion? And, if vampires exist, shouldn't we all be sucking blood by now? Students model the exponential growth of a contagion and use logarithms and finite geometric series to determine the time needed for a disease to infect the entire population.

Topic:
Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

Sign In

Like the jacket, this lesson is for Members only.

Mathalicious lessons provide teachers with an opportunity to teach standards-based math through real-world topics that students care about.

How do the rules of an election affect who wins? Students calculate (as a percent) how much of the electoral and popular vote different presidential candidates have received, and add with integers to explore elections under possible alternative voting systems.

Topic:
Number System (NS), Ratios and Proportional Relationships (RP), Reasoning with Equations and Inequalities (REI)