Volume and surface area are often compared by manufacturers in order to
maximize how much of something can go inside of a package (volume) while
keeping how much material is required to create the package (surface area) low.
Pick a product that might be packaged in the shape of a rectangular prism. A
rectangular prism has three dimensions: length, width, and height. The surface
area of a rectangular prism can be found using the formula SA = 2lw + 2wh + 2lh.
The volume of a rectangular prism can be found using the formula V = lwh. Write
an expression for the ratio of surface area to volume for the figure.
Choose an appropriate length, width, and height for your package so that it can fit
the product you are shipping. Using these dimensions, what is the ratio of surface
area to volume?
Product: Box of cereal
Dimensions: length = 8 inches, width = 3 inches, height = 11 inches
Surface Area: SA = 2lw + 2wh + 2lh = 2(8)(3) + 2(3)(11) + 2(8)(11) = 242 square inches
Volume: V = lwh = (8)(3)(11) = 264 cubic inches
Ratio of Surface Area to Volume: SA/V = 242/264 = 0.91
So, for this particular box of cereal, the ratio of surface area to volume is 0.91. This means that for every 1 unit of volume, we need 0.91 units of surface area to package the product. This ratio can be used by manufacturers to optimize their packaging strategy and reduce material costs while maximizing the amount of product that can be shipped in a single package.