Question

if the volume of an open box with a square base is 5 cubic feet, write the surface area, S, of the box as a function of x, the length of a side of the base?

Answer 1

Check the picture below.

the box is an open box, meaning the top is open, so that is not taking up area, so is not part of the surface area.

so can just get the area of the bottom, and the sides and sum them up, keeping in mind that the bottom is just an area of x² and the sides are just 4 rectangles of area h*x.

$\bf SA=\stackrel{\textit{bottom}}{(x\cdot x)}~~+~~\stackrel{\textit{4 lateral sides}}{4(hx)}\implies SA=\stackrel{\textit{bottom}}{x^2}~~+~~\stackrel{\textit{4 lateral sides}}{4\left(\cfrac{5}{x^2}\cdot x \right)} \\\\\\ SA=x^2~~+~~4\cdot \cfrac{5}{x}\implies SA=x^2+\cfrac{20}{x}$