A rectangula sheet of tin 45 cm by 24 cmis to be made into

Solution: Length and breadth of rectangular sheets 45 cm and 24 cm respectively.

Let side of each of the four squares cut off from each corner = x cm

Let Z denote the volume of the box

Vlume of box $(Z)=l.b.h$

$Z = (45-2x)(24-2x)x$

$\Rightarrow Z = (1080-138x +4x^2)x$

$\Rightarrow Z = 1080x -138x^2+4x^3$

Now differentiating with respect to x

$\frac{dZ}{dx} = 1080 - 276x +12x^2$

and $\frac{d^2Z}{dx^2} = -276 +24x$

$\text { For Max and minima } \frac{d z}{d x}=0$

$12 x^2-276 x+1080=0$

$=x^2-23 x+90=0$

$(x-5)(x-18)=0$

$x=5 \text { or } x=18$

$x=18$ is rejected because at $x=18$ length $=24-2 x=18-2 \times 18=-12$ which is impossible.

Here $x=5$ is the turning point.

$\text { At } x=5,$

$\frac{d^2 z}{d x^2}=24 \times 3-276=-156 \text { [Negative] }$

$z$ is maximum at $x=5$ that is, side of each square to be cut off from each corner for maximum volume is $5 \mathrm{~cm}$ .