Lesson 16

Minimizing Surface Area

16.1: The Least Material (5 minutes)

Warm-up

The purpose of this activity is to introduce students to the central problem of the lesson: how to calculate which cylinder takes the least material to build for a specific volume. Before students begin working with surface areas of cylinders, they are explicitly asked to make an estimate. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1).

Launch

Before students do calculations or any other work, display the image of the cylinders and poll the class about which option they think requires the least material to build. Display the results of the poll for all to see.

Student Facing

Here are four cylinders that have the same volume.

Four cylinders labeled A, B, C and D. Each cylinder has a different height and radius. A is tall with a small radius, D has the largest radius and is the shortest.
  1. Which cylinder needs the least material to build?
  2. What information would be useful to know to determine which cylinder takes the least material to build?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of this discussion is to make sure students remember how to calculate the volume and surface area of a cylinder.

Select students who chose A, B, C, and D to share their reasoning for why that cylinder requires the least material to build. If not brought up during the discussion, discuss how to calculate the volume of a cylinder \(V=\pi r^2h\) and the surface area of a cylinder \(S=2\pi r^2 +2\pi rh\) and write these formulas for all to see where they can remain displayed throughout the lesson. Consider using a rolled piece of paper to demonstrate the rectangular part of the surface area. If students are unsatisfied with not knowing which of the 4 has the smallest surface area, assure them that we will revisit the problem at the end of the lesson.

16.2: Figuring Out Cylinder Dimensions (15 minutes)

Activity

In this activity, students calculate the height needed for a cylinder with a specific volume and radius. The number of calculations needed to fill in the table is purposeful in order to encourage students to rearrange the formula for the volume of a cylinder to best suit their needs (MP8). The table and subsequent graph also help students identify that as one value increases (radius), the other decreases (height). The last question is meant to elicit student language about the general shape of the graph as well as allude to a topic that will be addressed directly in future lessons: horizontal asymptotes.

Monitor for students who rearrange the formula for the volume of a cylinder to share during the discussion.

Launch

Arrange students in groups of 2. Give groups 4–5 minutes of work time and then pause the class, selecting 1–2 previously identified students to share how they did the calculations for height by rearranging the formula for the volume of a cylinder as \(h=\frac{\text{volume}}{\pi r^2}\) and record this for all to see next to the equations recorded from the previous activity.

Provide access to devices that can run Desmos or other graphing technology.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Pour liquids from different-sized cylinders. Encourage students to recognize that, for containers with similar volumes, if the height decreases, the radius increases, and vice versa. This is also a good place to discuss how our perceptions can fool us: do these containers really look like they have the same volume?
Supports accessibility for: Conceptual processing

Student Facing

There are many cylinders with volume 452 cm3. Let \(r\) represent the radius and \(h\) represent the height of these cylinders in centimeters.

  1. Complete the table.

    volume (cm3) radius (cm) height (cm)
    452 1  
    452 2  
    452 3  
    452 4  
    452 5  
    452 6  
    452 7  
    452 8  
    452 9  
    452 10  
    452 \(r\)  
  2. Use graphing technology to plot the pairs \((r,h)\) from the table on the coordinate plane.

  3. What do you notice about the graph?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may mistakenly interpret \(\pi r^2\) in the volume formula as \((\pi r)^2\). Remind these students that, without parentheses to include \(\pi\), the squaring only applies to \(r\).

Activity Synthesis

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 \((r,h)\) 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 \(r\) was very, very small?” (If \(r\) is very, very small, then \(h\) would be very, very big, so the point would be close to the \(h\) axis and very far from the \(r\) axis in the positive direction.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share how they calculated the height of the cylinders given different radii, present an incorrect calculation and justification. For example, “The height of the cylinder with radius 2 is 11.4 because \(452\div(pi\boldcdot2)^2=11.4\).” Ask students to identify the error, critique the reasoning, and write to the author explaining the error. As students discuss with a partner, listen for students who clarify the steps to rearrange the volume formula. Invite students to share their critiques and explanations with the class. For example, a student may write, “when rearranging the formula, you must keep the exponents with the original terms” or “In the original formula, only the radius is squared, not the radius and \(pi\).” This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify how to manipulate volume formulas and consider the relationship between the height and radius of a cylinder.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

16.3: Calculating Surface Area (15 minutes)

Activity

The purpose of this activity is for students to combine formulas together to create an equation for the function relating the radius and surface area of cylinders with a specific volume, and graph it to learn more about it. Students also learn that the relationship they are investigating is called a rational function along with some features of equations of rational functions.

Launch

Arrange students in groups of 2. Tell half of the groups to calculate the surface area of a cylinder with radius 2 cm and the other half to calculate the surface area of a cylinder with radius 3 cm and to put their calculations into the table. Give groups 3–5 minutes of work time, and then select groups to share their calculations. It turns out that the surface area when \(r=3\) is smaller than when \(r=2\). Tell students to discuss a strategy with their partner to figure out the radius that leads to the smallest surface area for the cylinder and then use the table to record their work. Provide access to devices that can run Desmos or other graphing technology.

Speaking: MLR8 Discussion Supports. To support students in producing statements about the relationship between the radius, height, and surface area of cylinders, provide sentence frames for students to use. For example, “It looks like . . . .”, “Why do you think . . . .”, and “Where does _____ show . . .?”
Design Principle(s): Support sense-making
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Using a can or paper cutouts as models, show the 3 separate parts of surface area. Show why the dimensions of the rectangular part are \(2 \pi r\) and \(h\). Ask students to draw a reference model of the 3 parts of the surface area. Highlight and annotate the drawings and the formula so students can connect the representations of the formula \(S= 2 \pi r^2 + 2\pi r h\).
Supports accessibility for: Conceptual processing

Student Facing

There are many cylinders with volume 452 cm3. Let \(r\) represent the radius of these cylinders, \(h\) represent the height, and \(S\) represent the surface area.

  1. Use the table to explore how the value of \(r\) affects the surface area of the cylinder.

    radius (cm) height (cm) surface area (cm2)
         
         
         
         
         
         
         
         
  2. Use graphing technology to plot the pairs \((r, S)\) on the coordinate plane.
  3. What do you notice about the graph?
  4. Write an equation for \(S\) as a function of \(r\) when the volume of the cylinder is 452 cm3.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

We can model a standard 12 ounce soda can as a cylinder with a volume of 410.5 cubic centimeters, a height of about 12 centimeters and a radius of about 3.3 centimeters.

  1. How do its dimensions compare to a cylindrical can with the same volume and a minimum surface area?
  2. What other considerations do manufacturers have when deciding on the dimensions of the cans, besides minimizing the amount of material used?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are unsure of how to write \(S\) as a function of \(r\), it may help them to start with the equation for surface area. This equation involves \(h\), which will need to be eliminated in order to find an equivalent expression that only involves \(S\) and \(r\). Remind students that in the previous activity, they expressed \(h\) in terms of \(r\), and that this can help them write \(S\) in terms of \(r\).

Activity Synthesis

The goal of this discussion is to name the relationship students have investigated in this lesson and identify the dimensions of the cylinder with volume 452 cm3 that has the smallest surface area, and so takes the least amount of materials to make.

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 8.2 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 326.1 cm2. Less surface area means less material used to make the cylinder, and that could make the cylinder cheaper to build for, say, a can manufacturer.)
  • “How did you figure out the equation for the relationship between \(S\) and \(r\)?” (We already knew that \(h=\frac{452}{\pi r^2}\) for these cylinders, so replacing the \(h\) values in the equation for surface area gives a new equation with only \(r\) and \(S\).)

Tell students that the relationship 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 \(S=2\pi r^2+\frac{904}{r}\) as \(S=\frac{2\pi r^3 + 904}{r}\), which looks more like a polynomial divided by a polynomial.

Lesson Synthesis

Lesson Synthesis

Display the image from the warm up for all to see, along with the following dimensions:

  • Cylinder A: \(r=2\), \(h=36\)
  • Cylinder B: \(r=3\), \(h=16\)
  • Cylinder C: \(r=4\), \(h=9\)
  • Cylinder D: \(r=6\), \(h=4\)

Tell students to use what they have learned to identify the cylinder with the smallest surface area (C). After work time, poll the class again and invite students who voted for different cylinders to share their reasoning. Work to reach consensus that cylinder C has the smallest surface area.

16.4: Cool-down - Minimum Materials (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Some relationships cannot be described by polynomial functions. For example, let’s think about the relationship between the radius \(r\), in centimeters, and the surface area \(S\), in square centimeters, of the set of cylinders with a volume of 330 cm3 (this is a volume of 330 ml). What radius would result in the cylinder with the minimum surface area?

We know these formulas are true for all cylinders with radius \(r\), height \(h\), surface area \(S\), and volume \(V\):

\(\displaystyle S=2\pi r^2 +2\pi rh\)

\(\displaystyle V=\pi r^2h\)

Since we are only interested in cylinders with a volume of 330 cm3, we can use the volume formula to rewrite the surface area formula as:

\(\displaystyle S=2\pi r^2+\frac{660}{r}\)

Can you see how? By using the volume formula rearranged as \(h=\frac{330}{\pi r^2}\) and then substituting \(\frac{330}{\pi r^2}\) for \(h\) in the formula for surface area.

We now have an equation giving \(S\) as a function of \(r\) for cylinders with a volume of 330 cm3. From the graph of \(S\) shown here, we can quickly identify that a radius of about 3.75 cm results in a cylinder with minimum surface area and a volume of 330 cm3.​​​​​

\(S\) is an example of a rational function. Rational functions are fractions with polynomials in the numerator and denominator. Polynomial functions are a type of rational function with 1 in the denominator.

Graph of a rational function with a point, origin O.

​​​​​​

In this situation, the height of a cylinder with fixed volume varies inversely with the square of the radius, \(h=\frac{330}{\pi r^2}\), which means that as the value of \(r^2\) increases, the value of the height decreases, and vice versa. In later lessons, we’ll learn more about different features of rational functions, like why their graphs can look like they are made of two separate curves.