# Thread: Dimensions of Box if a Minimum?

1. ## Dimensions of Box if a Minimum?

This is the question:
"A new fruit juice (like a Popper) is to be marketed in a new container in the shape of a rectangular prism. The cardboard container is to have a square base and is to contain 300 mL of juice. What should be the dimensions of the container be if the amount of cardboard used in its construction is to be a minimum? Disregard waste and overlap."

All I have so far is:
Volume = l x w x h
Therefore:
300 = l x w x h
Therefore:
l=300/wh
I also know that the Surface Area of a prism is: 2(wh+lw+lh).
I have subbed l=300/wh into the SA equation and ended up with:
SA = 2wh + 600/h + 600/w

I honestly have no idea where to go from here (is what I've done correct?)...
Do I eventually have to derive an equation and then let that = 0 and then factorise?
I would really appreciate some help, guys.
Thank you in advance!

2. ## Re: Dimensions of Box if a Minimum?

It has a square base so l=w
$\displaystyle 300= w^2h$

Surface area= $\displaystyle 2(w^2+2hw)$

If you have done other Area/Volume optimisation problems you should be able to solve it form here

3. ## Re: Dimensions of Box if a Minimum?

How do you know it has a square base?

4. ## Re: Dimensions of Box if a Minimum?

Because you said so!
"The cardboard container is to have a square base".

5. ## Re: Dimensions of Box if a Minimum?

OMG! Thanks! Made things so much easier!