Question

A square piece of tin of side $24\, cm$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum?

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (B)

Let $x$ cm be the length of a side of the square which is cut-off from each corner of the plate. Then, sides of the box as shown in figure are $24 - 2x, 24 - 2x$ and $x$.
Let $ V$ be the volume of the box. Then,
$V = (24 - 2x) ^2 x = 4\,x ^3 - 96\,x ^2 + 576\,x$
$\Rightarrow \frac{dV}{dx} = 12x^{2} -192x + 576$ and $\frac{d^{2}V}{dx^{2}}$
$ = 24x - 192$
For maximum or minimum values of $ V $, we must have
$\frac{dV}{dx} = 0$

$ \Rightarrow 12 x^{2} - 192 x + 576 = 0$
$ \Rightarrow x^{2} -16 x + 48 =0$
$ \Rightarrow \left(x-12\right)\left(x-4\right) = 0$
$ \Rightarrow x= 12, 4$
But, $x= 12$ is not possible
Therefore, $x=4$.
and $\left[ \frac{d^{2}V}{dx^{2}}\right]_{x=4} = -96 <0 $
Hence, volume is maximum when $x = 4$.