the open box problem

**Brian** · 02-27-2003

lo.
Being a general retard, I am having difficulty with my coursework, and the deadline grows ever nearer.

The problem stated is:

An open box is to be made from a sheet of card
Identical squares are cut off the four corners of the card as shown in Figure 1

Code:

____________________
|_|...............|_|
| .               . |
| .               . |
| .               . |
| .    Figure 1   . |
| .               . |
| .               . |
|_................._|
|_|_______________|_|

The card is then folded along the dotted lines to make a box.

The main aim of this activity is to determine the size of the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card.

Part one
For any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

Part two
For any sized rectangular sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

You may do practical work and/or work in symbols.
Your teacher will help you choose the best way of working on the problem

I'm not expecting you to do my homework for me, but it would be extremely helpful if you could point me in the right direction.

I have done:
I have figured out the formula for working out the volume for the dimensions:
cutoffwidth * (lengthofcard - 2 * cutoffwidth) * (widthofcard - 2 *cutoffwidth)

and I have discussed how to calculate the formula for finding a square piece of cards' optimal cutoff width from it's dimensions, which is:
Optimal Cutoff Width of Square = Square's Width (or length) / 6

being mentally handicapped when it comes to mathematics, I am stuck on the bit which involves working out how to find the optimal cutoff width for a rectangle. The only method I have been able to employ so far is brute force.

Hope at least one of you has understood that.
Furthermore, I hope that person can help.

Thanks in advance.

**golfinguy4** · 02-27-2003

Are you in calc?

EDIT: I don't feel like waiting around so I'll give you the calc way of doing it which can be modified for other courses. You'd want to take the derivative of the volume function.

V=(l-2s)*(w-2s)*s

where l=length of piece, w=width of piece, s=length of square cut out

From here, expand and get
V=4s^3 - 2ls^2 - 2ws^2 + lws

Now, take the derivative of V so that you can find the maximum of V (If you aren't in calc, you are going to have to find the max graphically).

**FillYourBrain** · 02-27-2003

this is a question of a maximum (extrema? I don't remember what they called it) Your function of volume is a function of the length of a side of the cut out boxes. simply calculate the maximum on the graph.

there's probably a more basic way but I know this can be done with a little basic calc. simply derive and find out where the f'(x) is 0.

**PJYelton** · 02-27-2003

if w=width of card, l=length of card, and x is the size of the box you are cutting out, then this is the equation:

x*(l-2x)*(w-2x)=y

To solve, you can graph this equation with l and w being whatever value you want, and your answer is where the graph is maximum and positive. The other way (and its been awhile since I've done this so I might be a little wrong) is to take the derivitive of the above equation for any value of w and l, set it equal to zero, and solve for x, and this should be your maximum as well.

**Brian** · 02-27-2003

Originally posted by golfinguy4
Are you in calc?

No, I'm in England.

Don't know what calc would be.

Thanks for the help so far. A friend of mine will be helping me with this, for now I'll just plot a few meaningless graphs to make it look like I've done work, and I might see a pattern in the mean time.

**hk_mp5kpdw** · 02-27-2003

This website has a bunch of problems dealing with these types of Min/Max problems. You should be able to get an idea of what you need to do by looking there.

duhman · 02-27-2003

duh ! how did you cook up that name 'hk_mp5kpdw' ?

**golfinguy4** · 02-27-2003

Calc would be calculus.

**hk_mp5kpdw** · 02-28-2003

duh ! how did you cook up that name 'hk_mp5kpdw' ?

Click here!

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