The length of the side of a square sheet of cardboard is 12 cm - Leaving Cert Mathematics - Question b - 2014

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

$V = \text{Length} \times \text{Width} \times \text{Height}$

Substituting the expressions for the length and width in terms of h:

$V = (12 - 2h)(12 - 2h)(h)$

Expanding this gives:

$V = (144 - 48h + 4h^2) h$

This simplifies to:

$V = 4h^3 - 48h^2 + 144h$

To find the maximum volume, we first take the derivative of the volume and set it equal to zero:

$\frac{dV}{dh} = 12h^2 - 96h + 144 = 0$

Solving for h:

• Divide the equation by 12:

  $h^2 - 8h + 12 = 0$

• Factor the quadratic equation:

  $(h - 2)(h - 6) = 0$

• Solutions give:

  $h = 2$ or $h = 6$.

Since h must be less than 6 (otherwise the width becomes zero), we have:

• $h = 2$ for maximum volume.

To find the maximum volume, substitute h back into the volume formula:

$V = 4h^3 - 48h^2 + 144h$

Substituting h = 2:

$V = 4(2)^3 - 48(2)^2 + 144(2)$

Calculating:

• $4(8) - 48(4) + 288$
• $32 - 192 + 288$
• $128$

Thus, the maximum volume of the box is 128 cm³.