IM Alg2.2.16 Lesson: Minimizing Surface Area

Here are four cylinders that have the same volume.

[img]data:image/png;base64,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[/img][br]Which cylinder needs the least material to build?[br]

What information would be useful to know to determine which cylinder takes the least material to build?[br]

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

Use graphing technology to plot the pairs (r,h) from the table on the coordinate plane.

What do you notice about the graph?

[size=150]There are many cylinders with volume [math]452cm^3[/math]. Let [math]r[/math] represent the radius of these cylinders, [math]h[/math] represent the height, and [math]S[/math] represent the surface area.[/size]

Use the table to explore how the value of r affects the surface area of the cylinder.

Use graphing technology to plot the pairs (r, S) on the coordinate plane.

What do you notice about the graph?[br]

Write an equation for [math]S[/math] as a function of [math]r[/math] when the volume of the cylinder is [math]452cm^3[/math].[br]

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.

How do its dimensions compare to a cylindrical can with the same volume and a minimum surface area?[br]

What other considerations do manufacturers have when deciding on the dimensions of the cans, besides minimizing the amount of material used?[br]