An open box with a square base is to be made out of a given quantity of cardboard of area 2 square units. Show that the maximum volume of the box is c³ / 6√3 cubic units.

To find the maximum volume of an open box with a square base made from a given area of cardboard, we start by defining the dimensions of the box. Let the side length of the square base be \( x \) and the height of the box be \( h \).

Setting Up the Problem

The surface area of the box consists of the base and the four sides. The area of the base is \( x^2 \), and the area of the four sides is \( 4xh \). Therefore, the total surface area can be expressed as:

Surface Area Equation:

x² + 4xh = 2

Expressing Height in Terms of Base Length

From the surface area equation, we can solve for \( h \):

4xh = 2 - x²

h = (2 - x²) / (4x)

Volume of the Box

The volume \( V \) of the box can be calculated using the formula:

V = x²h

Substituting the expression for \( h \) gives:

V = x² * (2 - x²) / (4x) = (2x - x³) / 4

Finding the Maximum Volume

To maximize the volume, we need to take the derivative of \( V \) with respect to \( x \) and set it to zero:

V' = (2 - 3x²) / 4

Setting V' = 0 leads to:

2 - 3x² = 0

3x² = 2

x² = 2/3

x = √(2/3)

Calculating the Height

Now, substituting \( x \) back into the equation for \( h \):

h = (2 - (2/3)) / (4√(2/3)) = (4/3) / (4√(2/3)) = 1 / √(2/3) = √(3/2)

Volume at Maximum Dimensions

Now we can find the maximum volume:

V = x²h = (2/3)(√(3/2)) = (2√6) / 6 = c³ / (6√3)

Final Result

Thus, the maximum volume of the box is:

Maximum Volume: c³ / 6√3 cubic units.