## Calculus 8th Edition

$s = w^{2} + 4$
The first step in solving this problem requires the volume formula for the box $v = lwh$ Since we know that the base is a square, we can set two of the sides equal to each other. $l = w$ This gives us $v = 2wh$ where $w$ is the side of our base. Plugging in the given $2m^{3}$ as our volume, we get: $2 = 2wh$ Solving it gives us: $h = \frac{2}{2w}$ $h = \frac{1}{w}$ Now since the question asks for the surface area of the open box in terms of a side of the base $w$, we must find the formula for the surface area of the box. Remember that the face opposite to the base isn't there because our box is open. This gives us the formula for the surface area of the box as $s = w^{2} + 4(h*w)$ $w^{2}$ is the area of the base and $hw$ is the area of one side. Since there are 4 sides besides the base and the top of the box, we multiplied $lw$ by $4$. Plugging in our $h = \frac{1}{w}$ that we derived from our volume formula, we get $s = w^{2} + 4((\frac{1}{w})*w)$ $s = w^{2} + 4(1)$ $s = w^{2} + 4$ as our final answer.