Question
A Company is evaluating two packaging options for its product line. The more efficient design will have the lesser ratio of surface area to volume. Should the company use packages that are cylinders or rectangular prisms ?
Hint:
Surface area of a cylinder: 
Surface area of a rectangular prism:
Volume of a cylinder: 
Volume of a rectangular prism: 
We are asked to find which packaging is more efficient, cylinder or prism.
The correct answer is: evident that the cylinder has lesser ratio of surface area to volume. Hence, it is the most efficient packaging
Step 1 of 3:
Let x be the side length of the prism and h be the height of the prism.
The volume of a rectangular prism of given by: , where w ,h and l are the width, height and length respectively.
Here, the length and width are same since it’s a square. So, the volume is:
The surface area is:
Their ratio is:
Step 2 of 3:
The volume of a cylinder is given by:
, where r and h are the radius and height of the cylinder respectively.
Here, the radius is same as the side length of the prism and the height is same as the height of the prism. So, we have;
The surface area is:
Their ratio is:
Step 3 of 3:
Comparing the ratios, (cylinder) and (rectangle). It is evident that the cylinder has lesser ratio of surface area to volume. Hence, it is the most efficient packaging.
Ratio between two values can be written as 
.
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