The Box Problem

• Use the Arithmetic Mean -- Geometric Mean Inequality to find the maximum volume of a box made from a 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sides to form a lidless box.
  
  

• Determine what shape boxes could be created by this method from the 25 by 25 square sheet to hold a volume of 200 cu. units. 400 cu units? 800 cu units?
  
  
• Generalize. Use the AM-GM Inequality to discuss the maximum volume of a box formed from an n X n square sheet of cardboard.
  
  
• Why will the AM-GM Inequality not be a useful tool when the sheet of cardboard is 20 by 25?