From each corner of a square piece of sheet metal 18 centimeters on a side, remove a small square and turn up the edges to form an open box. What should be the dimension of the box so as to maximize its volume?

Height = 3 cm, Length = 9 cm, Width = 9 cm      

Step-by-step explanation:

A square piece of sheet of side x centimeters is cut from the square sheet.  When it was folded to make the box the height of box becomes x, length becomes (18-2x) and the width becomes (18-2x).

Volume is given by  

First, we differentiate V(x) with respect to x, to get,

Equating the first derivative to zero, we get,

Solving, with the help of quadratic formula, we get,

Again differentiation V(x), with resopect to x, we get,

At x = 9

At x = 3

Thus, by double differentiation test, the maxima occurs at x = 3 for V(x).

Thus, largest volume the box can have occurs when x = 3.

Dimensions of box:

Height = x  = 3 cm

Length = (18-2x) = 9 cm

Width = (18-2x) = 9 cm