IM Alg2.2.16 Lesson: Minimizing Surface Area
Here are four cylinders that have the same volume.
Which cylinder needs the least material to build?
What information would be useful to know to determine which cylinder takes the least material to build?
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?
There are many cylinders with volume . Let represent the radius of these cylinders, represent the height, and represent the surface area.
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?
Write an equation for as a function of when the volume of the cylinder is .
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?
What other considerations do manufacturers have when deciding on the dimensions of the cans, besides minimizing the amount of material used?