Question

An open-top box is to be made from a 12-inch by 24-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What is the maximum possible volume, in cubic inches, of the box?

Answer 1

Answer:

The maximum possible volume = 332.685 cubic inches

Step-by-step explanation:

From the information given:

Let the height be = x

The length be = 24 - 2x

The width be = 12 - 2x

Then V = x (24 -2x) ( 12 - 2x)

V = x ( 288 -48x - 24x +4x²)

V = x(288 - 72x + 4x²)

V = 288x - 72x² + 4x³

$\dfrac{dV}{dx}= (3x^2 - 36x+ 72)$

$\implies x^2 - 12x + 24 = 0$

$=\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$=\dfrac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(24)}}{2(1)}$

$=\dfrac{12 \pm \sqrt{144 - 96}}{2}$

$=\dfrac{12 \pm \sqrt{48}}{2}$

where;

$x \ne \dfrac{12 + \sqrt{48}}{2}$

∴ $x= \dfrac{12 - \sqrt{48}}{2}$

x = 2.536 ( since the length cannot be negative)

So, the length x = 24 - 2(2.535) = 18.93

The width = 12 - 2(2.535) = 6.93

heigth = 2.536

∴

V = 18.93 ×  6.93 ×  2.536

V = 332.685 cubic inches