Answer: The volume of the box is given by the formula:
(a) Equation of the box volume:
[tex]\begin{gathered} V=l\times w\times h\rightarrow(1) \\ \\ l=\text{ Length} \\ \\ w=\text{ Width} \\ \\ h=\text{ Height} \end{gathered}[/tex]
According to the figure, we can rewrite the formula (1) as follows:
[tex]\begin{gathered} l=(10-2x) \\ \\ w=(8-2x) \\ \\ h=x \\ \\ V(x)=(10-2x)(8-2x)x\rightarrow(2) \end{gathered}[/tex](b) The value of x for which the volume is the greatest is:
[tex]\begin{gathered} \begin{equation*} V(x)=(10-2x)(8-2x)x \end{equation*} \\ \\ \frac{dV(x)}{dx}=0 \\ \\ \\ \frac{d}{dx}[(10-2x)(8-2x)x]=0\rightarrow(3) \end{gathered}[/tex]The solution to the equation (3) is as follows:
[tex]\begin{gathered} \begin{equation*} \frac{d}{dx}[(10-2x)(8-2x)x] \end{equation*} \\ \\ \\ \frac{d}{dx}[4x^3-36x^2+80x]=0 \\ \\ \\ 12x^2-72x+80=0 \\ \\ \therefore\rightarrow \\ \\ \\ x=1.472in \\ \\ \\ \\ x=1.5in \end{gathered}[/tex]