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# Canalysis

## What’s the ideal size of a soda can?

What’s the ideal size of a soda can? Soda companies spend billions of dollars each year to manufacture 12-ounce cans. If the companies changed the cans’ dimensions, though, they would save lots of money.

In this lesson, students create rational functions to explore the relationship between volume, surface area, and cost to determine the optimal size of a soda can.

### REAL WORLD TAKEAWAYS

• The cheapest version of a product won’t necessarily be the most profitable.
• Small tweaks in per unit expenses can result in huge savings of costs in mass manufacturing.

### MATH OBJECTIVES

• Write, graph, and solve rational equations to describe geometric relationships

Appropriate most times as students are developing conceptual understanding.
Algebra 2
Rational Functions
Algebra 2
Rational Functions
Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (a) Graph linear and quadratic functions and show intercepts, maxima, and minima. (b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (c) Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (d) (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (e) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Mathematical Practices MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.7 Look for and make use of structure.