Folded boxes (a). Squares with sides of ?length x ? are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. (b). Suppose that in part (a) the original piece of cardboard is a square with sides of ? length ??. Find the volume of the largest box that can be formed in this way. (c). Suppose that in part (a) the original piece of cardboard is a rectangle with sides of ? ?length ?? an? d? . Holding ?€ fixed, find the size of the c?orner squares x ? that maximizes the volume ? of the box as L ? ? ?. (?Source: Mathe? matics Teacher, November 2002)

Solution 26E Step 1: (a) onsider abcd is the rectangular sheet of paper such thatab andcd are the lengths and ac andbd are the widths. If a square with sides of length x is cut from each corner, then the new dimensions of the sheet measuring 3ft by 4ft will be in length and in width. Consider the following figure