#### An open rectangular box with volume 2m^{3} has a square base. Express the surface area of the box as a function of the length of a side of the base.

#### Solution:

Let “x” represent the length of the box and “h” represent the height of the box. We know that the volume of the box is 2m^{3}, therefore,

2=hx^{2}h=\frac{2}{x^{2}}

The surface area of the box is, S=x^{2}+4xh. If we write the surface areas a function of “x” then

S(x)=x^{2}+4x(\frac{2}{x^{2}})

S(x)=x^{2}+(\frac{8}{x}) and the domain is x>0.

Click here for diagram.

How can a unit of volume be m^2?? isn’t it supposed to be cubed?

We fixed the typo. Thank you very much!

Can you please explain why do you multiply by 4 here: S = x2 + 4xh

Hi! So this is an open rectangular box with a square base. So if you draw it out, you will see that the bottom side is simply “x*x” (square base, which means each side is equal in length) which gives you surface area of the bottom side. Then we have 4 sides only, since the top is open. Each of the 4 sides is h*x (height x length). Since we have 4 of them, we get 4xh. So if we add everything to get the total surface area, we get, x^2+4xh.

Hopefully that clears it up, if it didn’t, let us know, and we will draw a diagram to explain it even better. Many thanks and best of luck with your studies! 🙂

Can you draw a diagram to explain it better to me? I still don’t get it how u get x^2+4xh.

Thank you.

Of course. Please refer to this diagram: https://bit.ly/2O2xkV5

As you can see on the diagram, there is a square base, and there is

notop. The height of the rectangle is “h” and the length is “x”. Because it has a square base, the surface area for that side is x*x. Now, we consider the other 4 panels, labeled 1 to 4. Each of those has a surface area of x*h. There are 4 of them. So we get, 4xh. Hopefully, that helps clear it up. Let us know if you still need more clarification.Best of luck with your studies 🙂