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 ℓ. 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 ℓ and L. Holding ℓ fixed, find the size of the corner squares x that maximizes the volume of the box as L →∞.