Question

A box is constructed from a rectangular metal sheet of $21\text{cm}$ by $16\text{cm},$ by cutting equal squares of sides $x$ from the corners of the sheet and then turning up the projected portions. The value of $x$ for which 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 $x.$
$V=x(21-2x)(16-2x)=x(336-74x+4x^2)$
$V=4x^3+336x-74x^2\Rightarrow \frac{dV}{dx}=12x^2+336-148x$
$\frac{dV}{dx}=0\Rightarrow 4(3x^2-37x+84)=0$
$\therefore (x-3)(3x-28)=0$
$\Rightarrow x=3,x=\frac{28}{3}$
Now, ${\frac{d^{2}V}{d{x}^2}|}_{x=3}=4(6x-37)=4(13-27)=-76<0\text{and}{\frac{d^{2}V}{d{x}^2}|}_{x=\frac{28}{3}}=4(56-17)=76>0$
Hence, $V$ is maximum when $x=3$