Question

A box, constructed from a rectangular metal sheet, is $21$ cm by $16$ cm by cutting equal squares of side $x$ cm from the corners of the sheet and then turning up the projected portions. The value of $x$ (in cm) so that volume of the box is maximum is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

The dimensions of the box after cutting equal squares of side $x$ on the corner will be $21-2x,16-2x$, and height $x$.
$V=x(21-2x)(16-2x)$
$=x(336-74x+4x^2)=4x^3+336x-74x^2$

$\therefore \frac{dV}{dx}=12x^2+336-148x$
$\Rightarrow \frac{dV}{dx}=0\Rightarrow 12x^2+336-148x=0\Rightarrow 3x^2-37x+84=0\Rightarrow (x-3)(3x-28)=0\Rightarrow x=3,\frac{28}{3}$
For $x=\frac{28}{3}\Rightarrow 16-2x<0$
Which is not possible,
$\therefore x=3$
$\frac{d^{2}V}{d{x}^2}=6x-37\Rightarrow \frac{d^{2}V}{d{x}^2}{|}_{x=3}=<0$
So for $x=3$ will give maximum volume.