A designer is creating various open-top boxes by cutting four equally-sized squares from the corners of a standard sheet of 8.5 inch by 11 inch paper, and then folding up and securing the resulting 'flaps' to be the sides of the box.

Let `x` represent the varying side length of the square cutouts in inches. Let `l,w,` and `h` represent the varying length, width and height of the box (in inches), respectively. Note that the width and length dimensions are such that `w < l` . Let `V` represent the varying volume of the box in cubic inches.

  1. Write formulas for length, width, and volume of the box, each in terms of `x` .

  2. Given the following starting and ending values of `x` , find the change in `x` and the resulting change in `l` .
    `x` changes from ... `Delta x`  `Delta l` 
    0 to 0.5 in
    0.5 to 1.5 in
    1 to 4 in

    Does `l` change at a constant rate with respect to `x` ? If yes, enter the appropriate value to complete the statement. If no, enter "DNE".
    `Delta l=` `* Delta x` .