Question

The World Cereal Organization (WCO) has asked you to redesign cereal boxes to more efficiently use paper. Assume the box must be rectangular with a square base, and that the top and the base are exactly the same. There are a total of 6 sides on the box. Use calculus to find the dimensions of the box that minimizes surface area (paper) while still holding 80 in3 of cereal. Your decision will dictate what the WCO implements, so you must justify your answer.

Answer 1

The length and breadth of the rectangular box with square base be x and its height be y. Then, the volume be given by, V=x​​​​​​2​​​ y=80, we get y= 80/x​​​​​​2 such that the total surface area is given by, S=2x2+4xy= 2x2+4x(80/x​​​​​​2). By finding the critical point and using the second derivative test, we will get the required Dimensions.