An open box with a square base is to be made out of a given quantity of card board of area `c^2` square units. Show that the maximum volume of the box is `(c^3)/(6sqrt(3))` cubic units.

Explanation

To solve the problem of maximizing the volume of an open box with a square base made from a given area of cardboard, we will follow these steps:

### Step 1: Define the Variables
Let the side length of the square base be \( a \) and the height of the box be \( b \). The area of the cardboard used is given as \( c^2 \).

### Step 2: Write the Surface Area Equation
The surface area \( S \) of the open box can be expressed as:
\[
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