A rectangular piece of zinc, 18 cm long and 12 cm wide, was used to make a rectangular box with no cover. How high should the box be to make it the maximum size?

To find the maximum size of the box, we need to maximize its volume.

Let's assume the height of the box is h cm.

The volume of the box is given by the formula:

V = l \cdot w \cdot h

where V is the volume, l is the length, w is the width, and h is the height.

Substituting the given values, we have:

V = 18 \cdot 12 \cdot h

To maximize the volume, we need to find the value of h that will give us the maximum result.

Taking the derivative of the volume function with respect to h, we have:

\frac{dV}{dh} = 18 \cdot 12

Setting the derivative equal to zero to find the critical point, we have:

18 \cdot 12 = 0

Simplifying, we get:

216 = 0

Since this is not possible, there are no critical points.

Therefore, to maximize the volume, we can choose any value for h as long as it satisfies the given conditions.

Answer: The height of the box can be any positive value.