How to Determine the Optimal Dimensions of a Tin Box to Minimize Material Use?

[jzq]

Minimizing Construction Costs: If an open box has a square base and a volume of 108 in.^3, and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction.

This is what I have so far:

$$Volume: 4y^3-4xy^2+x^2y=1=108$$

$$x=-\sqrt{\frac{108}{y}}+2y$$

Now I'm not sure if these are right so, please feel free to correct me. I would much appreciate it!

 

[dextercioby]

Who's "x",who's "y" and what is the shape of the box...?The simplest would be the case of a rectangular parallelipiped.


Daniel.

 

[Jameson]

Optimization problems like these deal with derivatives. I don't quite understand the question though. Is it saying you have an square box with an open top?

 

[jzq]

I copied this question write out of the book. It's hard to interpret. I'm thinking it has a square base, however the height varies. Therefore I put x-2y as the width and length and y as the height. The box is made from a tin sheet, so I believe the corners are cutout and folded up to form a box. That is why the sides are x-2y and the height is y. This is how I got the volume formula.

 

[dextercioby]

I hope they meant an parallelipiped.So what's the volume function...?It should be:
$$V=abc=108$$,okay...?

No make use of the fact that the base is a square,which means
$$V=a^{2}c=108$$

What's the area...?Remember,the area of the top must be taken out...


Daniel.

 

[jzq]

I hope they meant an parallelipiped.So what's the volume function...?It should be:
$$V=abc=108$$,okay...?

No make use of the fact that the base is a square,which means
$$V=a^{2}c=108$$

What's the area...?Remember,the area of the top must be taken out...


Daniel.

Sorry, I have no idea what you mean by parallelpiped. I wish I can draw a picture and upload it, but unfortunately the files are too large.

 

[dextercioby]

Can't u zip (turn to archive .zip) it or put it a .gif or .jpeg ?

Paralellipiped is a regular prism with all faces parallelograms...The natural generalization of a cube.

Daniel.

 

[Jameson]

He's saying you have a square base with a long/shorter height that isn't equivalent to the length of width of the base.

V for box like shape is abc. Since you have a * a * c(height), you can write this like Daniel did above: $V = a^2c$

This would give you the volume of the enitre box. But since we are picturing this figure to have the top removed, you need to subtract the area of the top to get an equation you can work with.

So use this equation:

$$V = a^2c - a^2$$

Does that make sense?


Jameson

 

[jzq]

Thanks a lot everyone!