Answer:
The surface area of the box as a function of the length of a side of the base is [tex]S=x^2+\frac{36}{x}[/tex].
Step-by-step explanation:
Let [tex]x[/tex] be the length side of the square base and [tex]y[/tex] be the height of the box.
According with the diagram the volume of a rectangular box with a square base is given by
[tex]V=x^2\cdot y[/tex]
and we know that its value is 9 [tex]m^3[/tex].
The surface area is given by
S = (Area of Base) + 4(Area of a Vertical Side)
[tex]S=x^2+4\cdot x\cdot y[/tex]
From the volume formula we can solve for [tex]y[/tex]
[tex]9=x^2\cdot y\\y=\frac{9}{x^2}[/tex]
and substitute into the surface area formula
[tex]S=x^2+4\cdot x\cdot \frac{9}{x^2} \\S=x^2+\frac{36}{x}[/tex]
