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

In summary, the conversation discusses finding the dimensions of an open box with a square base and a volume of 108 in.^3, constructed from a tin sheet with a minimum amount of material. The speakers discuss using a volume formula of V = a^2c - a^2 and the concept of a parallelipiped. They also mention the potential use of derivatives in optimization problems.
  • #1
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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:

[tex]Volume: 4y^3-4xy^2+x^2y=1=108[/tex]

[tex]x=-\sqrt{\frac{108}{y}}+2y[/tex]

Now I'm not sure if these are right so, please feel free to correct me. I would much appreciate it!
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
I hope they meant an parallelipiped.So what's the volume function...?It should be:
[tex] V=abc=108 [/tex],okay...?

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

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


Daniel.
 
  • #6
dextercioby said:
I hope they meant an parallelipiped.So what's the volume function...?It should be:
[tex] V=abc=108 [/tex],okay...?

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

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.
 
  • #7
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.
 
  • #8
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: [itex]V = a^2c[/itex]

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:

[tex] V = a^2c - a^2[/tex]

Does that make sense?


Jameson
 
  • #9
Thanks a lot everyone!
 

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

What is optimization?

Optimization is the process of finding the best possible solution to a problem or situation. It involves maximizing or minimizing a certain objective or criteria, while considering various constraints.

What are some common types of optimization problems?

Some common types of optimization problems include linear programming, nonlinear programming, integer programming, and dynamic programming. These problems can be applied to a variety of fields, such as engineering, economics, and computer science.

How do you approach an optimization problem?

The first step is to clearly define the objective and constraints of the problem. Then, you can use various mathematical techniques and algorithms to find the optimal solution. This may involve formulating a mathematical model, analyzing the problem to identify key variables, and using optimization software to solve the problem.

What is the difference between local and global optimization?

In local optimization, the objective is to find the best solution within a specific region or range. This may not necessarily be the globally optimal solution. In global optimization, the objective is to find the best possible solution over the entire feasible region, which may require more computational resources.

How is optimization used in real-world applications?

Optimization is used in a wide range of real-world applications, including supply chain management, financial planning, resource allocation, and machine learning. It helps to improve efficiency, reduce costs, and make data-driven decisions in various industries and disciplines.

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