A square board of side 36cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

A square board of side 36 cm is made into a box without top by cutting a square from each corner and folding up the flaps.

Let the side of the square cut from each corner be $x$ cm.

After cutting and folding:

Height of box $= x$ cm

Base of box $= (36 - 2x)$ cm $\times$ $(36 - 2x)$ cm

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Volume of the box:

$V = (36 - 2x)(36 - 2x)(x)$

$V = (36 - 2x)^2 \cdot x$

$V = (1296 - 144x + 4x^2) \cdot x$

$V = 1296x - 144x^2 + 4x^3$

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

$\dfrac{dV}{dx} = 1296 - 288x + 12x^2$

Setting $\dfrac{dV}{dx} = 0$:

$12x^2 - 288x + 1296 = 0$

$x^2 - 24x + 108 = 0$

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Using the quadratic formula:

$x = \dfrac{24 \pm \sqrt{24^2 - 4(1)(108)}}{2}$

$x = \dfrac{24 \pm \sqrt{576 - 432}}{2}$

$x = \dfrac{24 \pm \sqrt{144}}{2}$

$x = \dfrac{24 \pm 12}{2}$

$x = 18$ or $x = 6$

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For $x = 18$: Base $= 36 - 2(18) = 0$ cm, which is not possible.

For $x = 6$: Base $= 36 - 2(6) = 24$ cm, which is valid.

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Maximum volume at $x = 6$:

$V = (36 - 2 \times 6)^2 \times 6$

$V = (24)^2 \times 6$

$V = 576 \times 6$

$V = 3456$ cm³

Therefore, the maximum volume of the box is 3456 cm³.