Type 2 Box Discussion and Generalization

Sample Solution.

When the diagram in Figure 5 is folded along the dashed lines the height of the box will be 12 inches. Hence a 12 × 12 inch square must be cut from each corner of the rectangular piece of material from which we construct this box.  See Figure A (which is not drawn to scale).

 Figure A.

The width of the base of the box is 20 inches. Hence a 10 × 12 inch rectangle must be cut from each corner the material. See Figure B (which is not drawn to scale).

 Figure B.

It follows that we can label adjoin other labels to Figure B to indicate the length of the base of the box. These are shown in Figure C along with L and W for the rectangular piece of material from which Figure 5 was cut.

 Figure C.

It follows from Figure C that 20 = W - 2(12)-2(10) and 50 = L - 2(12). Hence the dimensions of the smallest rectangular piece of material that can be used to construct this box is W = 64 and L = 74.

A more general setting.

Consider a box of this type as shown in Figure D with volume V = xyz.

 Figure D.

Determine equations for the length L and width W of the smallest rectangular piece of material that can be used to construct this box.