Question

Cal 3
Need help getting this: design a closed rectangular box of minimum cost. suppose the box is to be of volume Vo cubic cm, and the cost of the material for the front and back sides is b dollars per square cm, c dollars per square cm for the left and right two sides, and d dollars per square for the top and bottom sides.

Answer 1

Like I understand a cost equation is v=xyz and c(xyz)=2bxz+2cyz+2dxz but I don't know how to go any further.

Answer 2

Are you doing Lagrange multipliers?

Answer 3

Yes we are

Answer 4

Why?

Answer 5

The problem needs to be set up using Lagrange multipliers.

Answer 6

You are trying to minimize
C(x,y,z)=2(bxy+cyz+dzx)
subject to the constraint:
\(xyz=V_0\)

Answer 7

Can you set up the standard equations using the Lagrange multiplier \(\lambda\)?

Answer 8

I will have to re read that section to re familiaze myself to do that. But once that's done does it just get answered through the use of multipliers?

Answer 9

It sounds difficult, but it is quite straight-forward.

The cost function to be minimized will be tagged on an extra term:
\(C(x,y,z)=2(bxy+cyz+dzx)+\lambda (xyz-V_0)\)
Do a partial differentiation with respect to each of the independent variables x,y, and z will give you 3 equations.
The fourth equation will be the constraint itself: \(xyz=V_0\).
The system (of 4 equations) will then be solved for x,y and z.

Note: you have a typo in the equation of the objective function. The first term should read 2bxy.

Answer 10

* forgot to mention: must equate each of the three partial derivatives to zero

Answer 11

* and the resulting system is non-linear. So either you solve it analytically, or you can use numerical (iterative) methods to find the values.

Answer 12

Okay. Thanks I'll try that this afternoon. Makes more sense....thanks for straightening me out and making it understandable.

Answer 13

Great, let me know if it works out.

Answer 14

Okay sorry been busy but what do you mean by analytically or iterative methods to solve? I just got back to working on this....been a busy week!

Answer 15

Have you set up the 4x4 equations?
The solution method could be quite simple.
Post your system of equations, and we can work together to solve it.

Answer 16

I messaged the teacher and he said I should be able to solve this with just the cost formula without the added section and with the Vo constant equation but what I don't understand is even with the equation Vo=xyz solved for say z and use that in the cost equation how does that work Vo is a constant but not a real number?

Answer 17

Well, your teacher is making use of symmetry, which is not a particular rigorous method.
However, it will (and should) give the same result if you went through with the Lagrange multiplier method.
Here's how it works.

Using the cost equation, we are trying to minimize
\(\large C(x,y,z)=2(bxy+cyz+dzx)\)
We also know that \(V_0\)=xyz, which can be substituted in C(x,y,z):
\(\large C(x,y,z)=2(b\frac{V_0}{z}+c\frac{V_0}{x}+d\frac{V_0}{y})\)
which simplifies to:
\(\large C(x,y,z)=2V_0(\frac{b}{z}+\frac{c}{x}+\frac{d}{y})\)
The minimum or maximum of C(x,y,z) occurs when the three quantities in parentheses are equal, meaning
\(\large \frac{b}{z}=\frac{c}{x}=\frac{d}{y}\)
are the conditions necessary for dimensions x,y,z to minimize C(x,y,z).
By the way, using Lagrange multipliers gives exactly the same result.

Answer 18

Okay but where's the actual numbers.....is it a guess a number or graph it and it will be a min or max?

Answer 19

There's nothing you can do until you have values for b,c, d and \(V_0\).
You can always create a numerical example by assigning numerical values to these variables.

Answer 20

Okay that's what I thought. Thanks.

Answer 21

Thank you for all the help but I'd just like to double check I understand this problem completely.
1)The only way to truly get a numerical answer is if I were to set a random number to the constant Vo?
2)Without any number set to the constant the best way to describe this problem to someone without math would be to describe the meaning of the C(x,y,z)=2Vo(b/z+c/x+d/y) and say in detail that in order to achieve the minimum cost would be to have the (variables) become equal?
3)How would you describe a case like b=2c=3d?

Answer 22

1) \(V_0\) does not have to be a random number, it should be a value prescribed by the question. You can select one to create an example for practice.
2. Again, as a test, you can set b=c=d=k and check if you have symmetry, i.e. x=y=z.
I totally agree that we are jumping to conclusions to say that b/z=c/x=d/y gives the max/min. I would have gone through the work of a Lagrange Multiplier calculation to arrive at the same conclusion.
3. b=2c=3d and b/z=c/x=d/y
implies
b/z=b/2x=b/3y
which in turn implies
z=2x=3y.

Answer 23

What do you mean a value prescribed by the question? The instructor/question doesn't have any value assigned to Vo

Answer 24

I don't know about the rules of your assignments.
Without further details, data or instructions, I would stop at the relation of the three variables as a function of the three constants, and supply the expression of C(x,y,z) as what you've got.

Answer 25

Recall that in math, the answer is not always a numerical value. Your job is done when you have given the method to design the box, while awaiting more numerical information.

Answer 26

Great. Thank you. With your help and little more from the instructor I finally understand this.

Answer 27

Now just to write it all up in a way that's understandable to someone who doesn't understand the math part shall be fun!

Answer 28

If you have completed the write-up, you are welcome to post it here, so I or someone else could critique it and share ideas.
Good luck with your work!