Question

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?

Answer 1

We have no dimensions to work with. I'll pick some and try and comply with the conditions of the problem. 

Suppose you have an object that is 14 by 22 by 27 cm. These three numbers have no common factor so they cannot be reduced any further, which is helpful for this problem.

Find the Volume
Volume

l = 27 cm
w = 14 cm
h = 22 cm

V = 27 *14 * 22
V = 8316 cm^3

Find the surface area
SA = 2*l*w + 2*l*h  + 2*w*h
SA = 2*27*14  +  2*27*22   +  2*14*22
SA = 756 + 1188 + 616
SA = 2558

Just looking at these numbers The surface area is about 1/3 of the volume. I don't think this is always true.


 $\frac{SA}{V} = \frac{2*L*W + 2*L*H + 2*W*H}{L*W*H}$ 

Another way to do this is to consider a cube which might give you a more useful result. 

s = L = W = H all three dimensions are equal in a cube.
The volume of a cube is s*s*s = s^3
The surface area of a cube is 2*s*s + 2*s*s + 2s*s = 6s^2

$\frac{SA_cube}{V_cube} = \frac{6s*s}{s*s*s}$
$\frac{SA}{V} = \frac{6}{s}$

That means whatever the side length, the Surface Area to volume = 6/the side length which is kind of an interesting result.