Express the surface of an open rectangular box with volume 3m^3 and a square base as a function of the length of the side of the base.

Answer 1

OK, so this is a 5-sided figure. The base is a square, which means that the 4 other sides will all have the same surface area (same length along the bottom and same height to the top)

Let's call the length of the side of the base "b".

1. Then, the surface area of the base of the box = b², so that's one side
2. The other 4 sides will all have the same surface area, A = b * h. However, we can't use "h" since the question just asks for a function of the side of the base.
3. But we do know that V = 3 m³, and we know that the volume of a rectangular box can be found with V = l *w * h, basically just multiplying all 3 sides together.
4. This shape is even easier, the volume is just the area of the base times the height
5. V = 3 m³ = b² * h
6. So we can solve that for h.
7. Then we can plug that expression for h back into our surface area formula for one of the sides, A = b * h
8. Now we put it all together
9. We have the area for one side of the box and we have the area for the base.
10. Remember that there are four sides, so multiply that area by 4, add it to the area of the base (b²), and you're done.