A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,

Let the side of the square to be cut off be x cm. Then, the height of the box is x, the length is 45 − 2x, and the breadth is 24 − 2x.

Therefore, the volume V(x) of the box is given by,

$\begin{aligned} V(x) &=x(45-2 x)(24-2 x) \\ &=x\left(1080-90 x-48 x+4 x^{2}\right) \\ &=4 x^{3}-138 x^{2}+1080 x \\ \therefore V^{\prime}(x) &=12 x^{2}-276 x+1080 \\ &=12\left(x^{2}-23 x+90\right) \\ &=12(x-18)(x-5) \\ V^{\prime \prime}(x) &=24 x-276=12(2 x-23) \end{aligned}$

Now, $V^{\prime}(x)=0 \Rightarrow x=18$ and $x=5$It is not possible to cut off a square of side 18 cm from each corner of the rectangular sheet. Thus, x cannot be equal to 18.

∴x = 5

Now, $V^{\prime \prime}(5)=12(10-23)=12(-13)=-156<0$

By second derivative test, x = 5 is the point of maxima.

Hence, the side of the square to be cut off to make the volume of the box maximum possible is 5 cm.