Question

A square piece of tin of side $24cm$ 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,
$V=(24-2x{)}^{2}x=4x^3-96x^2+576x$
$\Rightarrow \frac{dV}{dx}=12x^2-192x+576$ and $\frac{d^{2}V}{d{x}^2}$
$=24x-192$
For maximum or minimum values of $V$, we must have
$\frac{dV}{dx}=0$

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