Hello all,

Heres a classic math problem on finding maximum volume.

A rectangular box has to be made such that it has 6 faces ( i.e a completely closed box ).
For making this box we have paper of area X cm^2 ( that will form the 6 faces ) and a wire of length Y cm so that all the 12 edges of the box ( For sticking the edges and strengthening).
We have to find the maximum possible volume of the box that can be formed with the available material if it is said that it is compulsory to use all the length of the wire.

Eg:
X = 14 , Y = 20 ANS : 3 ( 3 , 1 , 1)
X = 16, Y = 20 ANS =4.15 (7/3,4/3,4/3)

The box has 3 dimensions, height, h. width, w, and length, l. The total area is 2hw+ 2lw+ 2lh= X. The total edge length is 4h+ 4w+ 4l= Y. You want to maximize volume, hwl, subject to those constraints.

The problem with not showing any attempt to do this problem is that we do not know what techniques you are familiar with. I would be inclined to use "Lagrange multipliers" but I don't know if you can do that.

The most "basic" method would be two use 2hw+ 2lw+ 2lh= X and 4h+ 4w+ 4l= Y to solve for, say, l and w in terms of h (and, of course, X and Y) and put those formulas into hwl so that you are maximizing a function of one variable.

Thank you for your reply.
Yes I am not really aware of lagarange's multiplier but if the problem cannot be solved without using it , I will understand it. The formula u have given above , I had come across over the internet but wasnt able to figure out how to exactly use them.
The thing is we have 2 equations and 3 unknowns , i m getting stuck there.
Can you please simulate with an example I have given ?

Thank you
Regards
Tanvi

But you are not solving equations. My point was that by using the given two equations, we can reduce the expression from three variable to one. You find possible max or min points by taking the derivative and setting it equal to 0.

I tried reducing it but 2 terms stickout , i m trying to understand lagaranges as well , so if a solution exists in that method also i will be glad if u could help me out !!
Infact theres a small clarification that , along with exact wire usage there should be exact paper area usage also !! I am sorry if this changes the thinking dimension to the problem !!

Thanks close topic ! it helped !!