(a).

Volume of the closed box with a square base is .

The volume and surface area  of a box with square base length  and height .

Therefore, .

(b).

(1). Draw the coordinate plane.

(2). Graph the function .

Graph :

(c).

The minimum amount of cardboard used corresponds to the graph of with the smallest

-coordinate.

Graph :

Observe the garph :

The graph has minimum point at .

Therefore, the minimum amount of cardboard used is and this occurs when .  

(d).

Consider .

Differentiate the above function with respect to .

Equate to zero.

The dimensions of the box that minimize the surface area are .

(e).

UPS is interested in designing a box that minimizes the surface area because to minimize the cost of materials used for the construction.

(a).

.

(b).

Graph :

(c).

Minimum amount of cardboard used is and this occurs when .

(d).

.

(e).

To minimize the cost of materials used for the construction