Question

A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 2 cm squares from each corner and folding up the sides. Let x represent the width of the original piece of cardboard. Find the width of the original piece of cardboard,x, if the volume of the box is 1120 cm^3

Answer 1

Let the width of the original cardboard piece be x cm. Then its length is 2x cm. After cutting corners which are 2cm squares, and after folding up the sides, an open box is formed with length 2x-4 cm , width x-4 cm and depth 2 cm.

The volume of the box is V(x) = length*width*depth = (2x-4)*(x-4)*2 = = 4x²-24x+32

Now, if the volume of the box is 1120 cubic cm then 4x²-24x+32 = 1120 or, x²-6x+8 =280 or, x²-6x -272 = 0. On using the quadratic formula, we have x = [ 6±√{36-4*(-272)}]/2 = (6±√1124)/2 =(6± 33.26)/2 . Now, since x cannot be negative , hence x = (6+33.526)/2 = 39.526/2 = 19.76

Thus, the width of the original cardboard piece is 19.76 cm. ( approximately, on rounding off to 2 decimal places).