Question

We have a $12\text{m}\times 12\text{m}$ piece of thin material and want to make an open box by cutting small squares at the corners of our material and folding the side up. Find out the side of small squares for which the volume of box becomes maximum.

Answer 1

Let,
$x$= length of cut on each sides of small squares
$V$= volume of folded box

The length of base after two cuts from each edge of size $x$ is $(12-2x)$ and depth of box after folding is $x$. Two equal length cut from $12\text{m}$ can each have a maximum length of $6\text{m}$. So the value of $x$ lies between 0 and 6 and at that values of $x$ volume is zero.

Volume of box can be written as
$V=x(12-2x)(12-2x)=4x^3-48x^2+144x$
To find out the maximum value we have to find the critical point,
$\frac{dV}{dx}=0$

$\frac{dV}{dx}=12x^2-96x+144=12(x-2)(x-6)=0$
$x=2,6$

There is only one value $x=2$ is possible at which $V=128m^3$ and this is the maximum volume of box
because the volume of box is zero at $x=6$.