QuestionGiven a $W=10$ inch by $L=13$ inch piece of paper, we will cut out squares (size $x$ by $x$ ) from each corner and fold to create an (open top) box. Our goal is to find the size of the cut out square $(x)$, that maximizes the volume of the box.
The length of the box, as a function of $\tau$, is $l=13-2 x \vee \sigma^{6}$
The width of the box, as a function of $x$, is $w=10-2 x \vee 0^{6}$
The volume of the box, as a function of $x$, is $V=13 \cdot 10 \cdot x \times 0^{6} x(13-2 x)(10-2 x)$ which, after distributing, simplifies to $V=130 x-46 x^{2}+4 x^{3} \quad o^{6}$ To determine the value of $x$ that corresponds to a maximum volume, we need to find $V^{\prime}$. $$V^{\prime}=130-92 x+12 x^{2}$$
The $\boldsymbol{x}$ that corresponds to a maximum volume is $\boldsymbol{x}=1.8712 \times 0$ ob 1.8683623129081 inches and the maximum volume is $V=$ $\square$ cubic inches