An open–top box is being designed by cutting a corner piece out of a 16" by 14" piece of metal and folding the sides upwards. the designer wants to maximize the volume of this box.

I hope you're in calculus doing this problem(?) If you cut away from both the length and the width, your sides have values then of 16 - 2x, and 14 - 2x, and the height is x.  Solving this gives you a cubic: 4x^3 - 60x^2 + 224x. If you take the derivative of this function, you will get where the max value of the function occurs, since that what the derivative tells you...either a max or a min value of a function. The derivative is 12x^2 - 120x + 224. If you use the quadratic formula and solve for x, you get 2 values: 7.5 and 2.48. Sub both those values back in to the original function to get that an x value of 7.5 plugged into the original gives you a -7.5. Well, we both know that volume cannot be a negative value, so sub in the x value of 2.48 to get the max volume of 247.507968 inches cubed.