Find the most economical dimensions of a closed rectangular box [..]

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  • #1
s3a
818
8
"Find the most economical dimensions of a closed rectangular box [. . .]"

Homework Statement


Find the most economical dimensions of a closed rectangular box of volume 8 cubic units if the cost of the material per square unit for (i) the top and bottom is 5, (ii) the front and back is 2 and (iii) the other two sides is 5.

(a) What is the length of the vertical edge?
(b) What is the length of the horizontal front and back edge?
(c) What is the length of the horizontal side edge?


Homework Equations


Lagrange Multiplier Method or equivalent.


The Attempt at a Solution


The Lagrange Multiplier method is not required for this problem so if there is an easier way please notify me of it as well as why it's easier. My work is attached but I'm stuck at the step of figuring out the variables/dimensions.

Any help would be GREATLY appreciated!
Thanks in advance!
 

Attachments

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  • #2


It would make more sense if you would specify what x, y, and z represent at the start.
I assume you mean that x and y are "length" and "width" and z is "height".

In any case, yes, the volume satifies xyz= 8 and the "cost" is 10xy+ 4xz+ 10yz.

You can write each of your erquations as
[tex]10y+ 4z= -\lambda yz[/tex]
[tex]10x+ 10z= -\lambda xz[/tex] and
[tex]4x+ 10z= -\lambda xy[/tex]

Since the value of [itex]\lambda[/itex] is not important to the solution, I like to eliminate it first by dividing one equation by another:
If you divide the first equation by the second you get
[tex]\frac{5y+ 2z}{5x+ 5z}= \frac{y}{x}[/tex]
and if you divide the first equation by the third you get
[tex]\frac{5x+ 2z}{2x+ 5z}= \frac{z}{x}[/tex]
Those, together with [itex]xyz= 8[/itex] give you three equations to solve for x, y, and z.
 
  • #3


s3a said:

Homework Statement


Find the most economical dimensions of a closed rectangular box of volume 8 cubic units if the cost of the material per square unit for (i) the top and bottom is 5, (ii) the front and back is 2 and (iii) the other two sides is 5.

(a) What is the length of the vertical edge?
(b) What is the length of the horizontal front and back edge?
(c) What is the length of the horizontal side edge?


Homework Equations


Lagrange Multiplier Method or equivalent.


The Attempt at a Solution


The Lagrange Multiplier method is not required for this problem so if there is an easier way please notify me of it as well as why it's easier. My work is attached but I'm stuck at the step of figuring out the variables/dimensions.

Any help would be GREATLY appreciated!
Thanks in advance!

Your attachment is so messy that it is unreadable; please be more careful when posting here.

RGV
 

FAQ: Find the most economical dimensions of a closed rectangular box [..]

1. What is the purpose of finding the most economical dimensions of a closed rectangular box?

Finding the most economical dimensions of a closed rectangular box allows us to minimize the cost of materials while still meeting the required volume or capacity of the box. This is important for businesses and manufacturers who want to reduce production costs.

2. How do you determine the most economical dimensions of a closed rectangular box?

The most economical dimensions of a closed rectangular box can be determined by taking into account the cost of materials, the required volume or capacity, and any other constraints such as weight or structural stability. This can be done through mathematical calculations or using optimization techniques.

3. What factors should be considered when determining the most economical dimensions of a closed rectangular box?

The factors that should be considered include the cost of materials, the required volume or capacity, any constraints such as weight or structural stability, and the overall efficiency of the design. It is also important to consider the potential for cost savings in the long run, such as reducing shipping costs or maximizing storage space.

4. Can the most economical dimensions of a closed rectangular box change over time?

Yes, the most economical dimensions of a closed rectangular box can change over time due to fluctuations in material prices, changes in production methods, or advancements in technology. It is important to regularly reassess and optimize the dimensions to ensure continued cost savings.

5. Are there any limitations to finding the most economical dimensions of a closed rectangular box?

Some limitations to consider include the availability of materials, the desired volume or capacity, and any other constraints such as weight or structural stability. Additionally, the cost of production may also need to be taken into account, as a cheaper material may require more labor or energy to produce, ultimately impacting the overall cost.

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