Answer: The volume of the box is given by the formula:
(a) Equation of the box volume:

According to the figure, we can rewrite the formula (1) as follows:

(b) The value of x for which the volume is the greatest is:
![\begin{gathered} \begin{equation*} V(x)=(10-2x)(8-2x)x \end{equation*} \\ \\ (dV(x))/(dx)=0 \\ \\ \\ (d)/(dx)[(10-2x)(8-2x)x]=0\rightarrow(3) \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/yrnfjox2rehwf5nmbf7ipt5tmcrxbfjo0j.png)
The solution to the equation (3) is as follows:
![\begin{gathered} \begin{equation*} (d)/(dx)[(10-2x)(8-2x)x] \end{equation*} \\ \\ \\ (d)/(dx)[4x^3-36x^2+80x]=0 \\ \\ \\ 12x^2-72x+80=0 \\ \\ \therefore\rightarrow \\ \\ \\ x=1.472in \\ \\ \\ \\ x=1.5in \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/ruerxna38euve2lim4d6s4eusmphnpihgv.png)