let the side of base =x and height=h
[tex]hx^2=16,823[/tex] so [tex]x=\sqrt{\frac{16,823}{h}}[/tex]
surface area of the box = S(x)= x² +4xh
to find the minimum we calculate the first derivative
S'(x) = 2x + 4h
this is minimum for S'(x) =0
2x + 4h = 0 replace x by its value..
[tex]2\sqrt{\frac{16,823}{h}}+4h=0[/tex]
[tex]4h=-2\sqrt{ \frac{16,823}{h} }[/tex] square both sides
[tex]16h^2=4\frac{16,823}{h}[/tex] cross multiply
[tex]h^3=4\frac{16,823}{16}[/tex]
so [tex]h=\sqrt[3]{\frac{16,823}{4}} =16.14 \text{ cm}[/tex]
