Question

A rectangular sheet of tin $45cm$ by $24cm$ is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?

Answer 1

let the side of square be $x$
Then remaining dimensions of cuboid for volume is
length$45-2x$, width $24-2x$ and height $x$
Volume(v)$=(45-2x)(24-2x)x\frac{dv}{dx}=(45-2x)(24-2x)-2x(24-2x)-2x(45-2x)\frac{dv}{dx}=12(x^2-23x+90)=12(x-5)(x-18)$
Now$\frac{dv}{dx}=0$ at $x=5,18$
Now,
$\frac{d^{2}v}{d{x}^2}=12(2x-23)$
at $x=5\frac{d^{2}v}{d{x}^2}<0\therefore atx=5$
that is side of square for which we will get maximum volume