A box with a square base....

Question

A box with a square base and no top is to be made from a square piece of carboard by cutting 7 in. squares from each corner and folding up the sides. The box is to hold 1792 in3. How big a piece of cardboard is needed?

Answer 1

Even if not drawn to scale, it helps to draw a picture:

 

        XXXXXXXXX   

        XXXXXXXXX

   XXXXXXXXXXXXXX

   XXXXXXXXXXXXXX

        XXXXXXXXX

        XXXXXXXXX

 

 

The volume of the box (made after folding) is:

      width*length*height

    (x-7-7)*(x-7-7)*7

    (x-14)(x-14)(7) = 1792

    (x-14)(x-14) = 256          [save time, divide by 7 now]

                  [also save time by realizing that 16*16=256, so x=30]

    x² - 14x -14x + 196 = 256

    x² - 28x - 60 = 0

    (x+2)(x-30) = 0             [either factor or use quadratic formula]

        x cannot be (-2)

        x = 30 in.

 

Check:

     Is    (30-7-7)(30-7-7)(7) = 1792    ?

              16*16*7 = 1792    ?               [note: the box dimensions]

                  1792 = 1792   ?yes