W. P. involving polynomial function

Question

The area of a rectangular piece of cardboard is 1008 cm^2. If you cut 4 cm sqaure from each corner of the rectangular piece of cardboard and turn up the sides then an open box is formed whose vol is 2240cm^3. Fiond the dimensions of the rectangular piece of cardboard and the dimension of the open box.

Answer 1

Let l = length of rectangular piece

     w = width of rectangular piece

 

 lw = 1008                          (1)

 

Noting that the base area of the 3D box is 2240/4=560cm² and that this area is the same as the "interior" rectangle of the original piece after the 4 corners have been cut (i.e. the part without the sides to be folded up):

 

(l-8)(w-8) = 560   (2)

 

Substitue  l = 1008/w from (1) into (2):

 

((1008/w) - 8)(w - 8) = 560

...

eventually boils down to

w²-64w+1008 = 0

 

w = (1/2)[64 ± √(64²-4(1)(1008)) ]

 

w = 32 ± 4  —>  w = 28 or 36

 

From (1), this means l = 36 or 28, respectively.

 

Choosing the smaller w and larger l:   original rectangle was 36cm x 28cm

 

and subtracting 8 from both l & w, the open box is:    28cm x 20cm x 4cm