First always draw a picture! Our figure having a square base as well as open top are both crucial pieces of information. When drawing your box, make sure to label length and with as the same variable, we'll call it x and the height will be different, call it y. We know V=lwh, but in this case l and w are both x and h is y so our volume formula will look like V=x2y. We know V=97556 so 97556=x2y. Next, we will want our surface area because that decides how much material we need to construct the faces and we want that minimized. SA is the sum of all the areas of the faces. Remember, the open top means we do not have a cap to this box. So let's find some areas.
Base-we have a square base with side x so area=x2
Left and right panels- these have dimensions x by y and there are two panels so area=2xy
Front and back panels- these also have dimensions x by y and there are two of them so area=2xy
Putting this all together we get SA=x^2+2xy+2xy=x^2+4xy
Recall our volume formula was 97556=x2y and we now have a surface area defined by SA=x^2+4xy
Let's solve for y in the volume formula since that would be easier than introducing square roots and get y=97556/x2.
Plugging into SA we get SA=x^2+4x(97556/x2)
An x will cancel giving us SA=x2+390224/x
Differentiating SA and setting it equal to 0, we get 0=2x-390224/x2
Multiplying by x2 we get 0=2x3-390224
Dividing by 2 we get 0=x3-195112
Solving for x we get x=58.
Now we have found x. However, we still need y. But we solved for this a while back when we isolated it in the volume formula. We had y=97556/x2
So, y=97556/(58)2
y=29
So we conclude length=width=x=58 and height=y=29
