Geometry cereal box

Question

Design a new cereal box that will hold the same amount of cereal but reduce manufacturing costs. Prove that your new design holds the same amount but can be manufactured more cheaply. 

*My box costs $11.6 

-So I have a rectangular prism thats 

8 in-length 

1 in-width

12 in-height

the volume is 96 in. and the surface area is 232 

*If you could show your work that would be great. 

Answer 1

A rectangular box of a given volume and minimum surface area is a cube, and the length, width, and height of a cube of volume 96 cubic inches is the cube root of 96, which is about 4.58" on a side. Since there are six squares making up the cube (top, bottom, front, back, left, right), the optimized surface area is six times

4.58 squared, which is less than 126 square inches, far less material (and thus less cost) than the box you started with. (An assumption here is that manufacturing cost is proportional to surface area.)