Minimize the Surface Area of a Cylinder

Many products are packaged in cylindrical containers. Consider the food items on the shelves in a grocery store. You can buy fruits, vegetables, soups, dairy products, potato chips, fish, and beverages in cylindrical containers.

Investigate

Tools

How can you compare the surface areas of cylinders with the same volume?

construction paper

ruler

Method 1: Build Models

scissors

tape

Your task is to construct three different cylinders with a volume of

500 cm3.

Work with a partner or in a small group.

1. Choose a radius measurement for your cylinder. Calculate the

area of the base.

2. Using the formula Vcylinder ϭ (area of base)(height), substitute

the volume and the area of the base. Solve for the height.

3. Calculate the circumference of the base.

4. Construct the rectangle that forms

the lateral surface area of the cylinder. The rectangle should have a length equal to the circumference you determined in step 3 and a width equal to the height you determined in step 2. Tape the rectangle to form the curved surface of the cylinder.

5. a) Calculate the area of the rectangle.

b) Calculate the total surface area of

the cylinder, including the base and the top.

510 MHR • Chapter 9

6. Record the results for this cylinder in a table.

Cylinder Radius (cm) Base Area (cm2) Height (cm) Surface Area (cm2)

1

2

3

7. Repeat steps 1 to 6 to create two different cylinders, each

with a volume of 500 cm3.

8. Compare the surface areas and dimensions of the cylinders.

Choose the cylinder that has the least surface area. How does its height compare to its diameter?

9. Reflect Compare your results with those of other groups in the

class. Describe the dimensions of the cylinder with the least surface area. Are these dimensions the optimal ones? Explain.

Method 2: Use a Spreadsheet

1. Use a spreadsheet to investigate the surface area of cylinders

with different radii that have a volume of 500 cm3. Start with a radius of 1 cm.

A

B

C

(cm2)

(cm3)

D

E

Height (cm)

Surface Area (cm2)

1

Radius (cm)

2

1

=PI()*A2^2

500

=C2/B2

=2*B2+2*PI()*A2*D2

3

2

=PI()*A3^2

500

=C3/B3

=2*B3+2*PI()*A3*D3

Base Area

Volume

4

2. Use Fill Down to complete the spreadsheet. What is the

whole-number radius value of the cylinder with the least volume? Try entering a radius value 0.1 cm greater than this value. Does the surface area decrease? If not, try a value

0.1 cm less. Continue investigating until the surface area is a minimum for the radius value in tenths of a centimetre.

3. What is the radius of the cylinder with minimum surface area?

How does this compare to the height of this cylinder?

4. Change the value of the volume in the spreadsheet to investigate

the dimensions of a cylinder with minimum surface area when the volume is 940 cm2. How do the radius and height compare?

5. Repeat step 4 for a cylinder with a volume of 1360 cm2.

6. Reflect Summarize your findings. Describe any relationship you

notice between the radius and height of a cylinder with minimum surface area for a given volume.

9.6 Minimize the Surface Area of a Cylinder • MHR 511

Example Minimize the Surface Area of a Cylinder

a) Determine the least amount of aluminum required to construct

a cylindrical can with a 1-L capacity, to the nearest square centimetre. b) Describe any assumptions you made.

Solution

a) For a given volume, the

cylinder with minimum surface area has a height equal to its diameter.

2r

2r

The front view of this cylinder is a square.

Substitute h ϭ 2r into the formula for the volume of a cylinder.

V ϭ r 2h ϭ r 2(2r) ϭ 2r 3

Substitute the volume of 1 L, or 1000 cm3, to find the dimensions of the cylinder.

1000 ϭ 2r 3

500

1

1000

2r 3 ϭ 2

2

˛˛˛

1

Divide both sides by 2. ç 500 ÷ π = ◊

1

3

500 ϭ r3

500 ϭr B и r

5.42 ϭ

3

Take the cube root of both sides.

The radius of the can should be 5.42 cm. The height is twice this value, or 10.84 cm.

To find the amount of