Some students may interpret in the volume formula as . If this happens, consider asking:

• “Can you explain how you calculated the heights in the table.”
• “What is the same and different about and ?”

Invite students to share things they noticed about the graphs they created. Here are some questions for discussion to prime students for the idea of asymptotes, which will be introduced in the next lesson.

• “Are there any cylinders for which the point could be on an axis?” (No, if one of the dimensions is 0, then the cylinder has no volume and does not exist.)
• “Where would the point be if the value of was very, very small?” (If is very, very small, then would be very, very big, so the point would be close to the -axis and very far from the -axis in the positive direction.)

If students are unsure of how to write as a function of , consider asking:

• “Can you explain how you calculated the surface areas in the table”
• “How could you write an equivalent equation with only and , no ?”

The goal of this discussion is to name the relationship students have investigated in this lesson and identify how to determine the dimensions of the cylinder with volume 452 cm³ that has the smallest surface area.

If time allows, pair groups to share their graphs and observations before the whole-class discussion. Otherwise, begin by inviting 2–3 groups to share their graphs and things they noticed about the graphs, recording responses for all to see. Here are some questions for discussion.

• “What values of radius and height result in the smallest surface area?” (A radius of about 4.2 cm with a height of 8.3 cm minimizes the surface area.)
• “What is the minimum surface area, and why would we care about finding this value?” (The minimum surface area is about 326 cm². Less surface area means less material used to make the cylinder, and less material could make the cylinder cheaper to build for, say, a can manufacturer.)
• “How did you figure out the equation for the relationship between and ?” (We already knew that for these cylinders, so replacing the -value in the equation for surface area gives a new equation with only and .)

Tell students that the relationships between the height and volume and between the surface area and radius of the cylinder are examples of rational functions. Rational functions include polynomials but allow fractions with polynomials in the numerator and denominator (so long as the denominator isn’t 0). If time allows, show students how to rewrite as , which looks more like a polynomial divided by a polynomial. Add the formula for in terms of to the list of formulas started earlier in the lesson.