Solution
Area of square base of sides x is
[tex]Area=x^2[/tex]Volume = 4000cm^3
[tex]\begin{gathered} Volume=Bh \\ B=Base\text{ }Area \\ h=height \end{gathered}[/tex]Thus,
[tex]\begin{gathered} Volume=Bh \\ 4000=x^2h \\ \\ h=\frac{4000}{x^2} \end{gathered}[/tex]For the box to require the least possible material, is to simply minimize the surface area of the rectangular box
The surface Area is given as
[tex]\begin{gathered} Area=2(lw+wh+lh) \\ Since,\text{ it is a square base} \\ l=x \\ w=x \\ \\ Area=2(x^2+xh+xh) \\ Area=2(x^2+2xh) \\ Area=2(x^2+2x(\frac{4000}{x^2})) \\ \\ Area=\frac{16000}{x}+2x^2 \end{gathered}[/tex]Now, we differentiate
[tex]\begin{gathered} Area=\frac{16,000}{x}+2x^{2} \\ A=16000x^{-1}+2x^2 \\ By\text{ differentiating} \\ \frac{dA}{dx}=-16000x^{-2}+4x \\ \\ At\text{ minimum area, }\frac{dA}{dx}=0 \\ 4x=16000x^{-2} \\ x^3=4000 \\ x=10\sqrt[3]{4} \end{gathered}[/tex]Now, to find h
[tex]\begin{gathered} h=\frac{4000}{x^2} \\ h=\frac{4000}{100(4)^{\frac{2}{3}}} \\ h=4^{\frac{1}{3}}\times10 \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} Length=10\sqrt[3]{4}cm=15.874cm\text{ \lparen to three decimal places\rparen} \\ Width=10\sqrt[3]{4}cm=15.874cm\text{ \lparen to three decimal places\rparen} \\ height=15.874cm\text{ \lparen to three decimal places\rparen} \end{gathered}[/tex]