Question

A square piece of tin of side $12cm$ is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum ? Also, find this maximum volume.

Answer 1

let the side of square be $x$
Then remaining dimensions of cuboid for volume is
length$18-2x$, width $18-2x$ and height $x$
Volume(v)$=(18-2x{)}^{2}x\frac{dv}{dx}=(18-2x{)}^2-2x(18-2x)\frac{dv}{dx}=(18-2x)(18-2x-2x)=(18-2x)(18-4x)$
Now$\frac{dv}{dx}=0$ at $x=9,\frac{9}{2}$
Now,
$\frac{d^{2}v}{d{x}^2}=-4(18-2x)-2(18-4x)$
at $x=\frac{9}{2}=4.5\frac{d^{2}v}{d{x}^2}<0\therefore atx=4.5$ that is side of square for which we will get maximum volume