Minimizing Total Cost of Making Open Box with Squared Base

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In summary, the problem involves finding the dimensions of an open box with a squared base that has a volume of 6400 cm^3 while minimizing the total cost of making the box. The cost is $1.00 /cm^2 for the base and $0.50/cm^2 for each of the four sides. By solving for y in the volume constraint and substituting it into the total cost function, the problem can be simplified to a function of only x, and then minimized to find the optimal dimensions. Lagrange multipliers may not be necessary for this problem.
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jetoso
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Homework Statement


We want to make an open box with squared base. Let x be the dimension of one side of the base and y the height of the box. We pay a cost of $1.00 /cm^2 for the base and $0.50/cm^2 for each side.
The box must have a volume V = 6400 cm^3. Determine the dimensions x and y which will minimize the total cost of making the box.


Homework Equations


V = yx^2, TC = x^2 + 0.5xy


The Attempt at a Solution


I tried to solve the problem taking the partial derivatives with respect to x and y, which will give me the minimum values, but my results are not consistent. I am thinking on using Lagrange multipliers to solve the nonlinear minimization problem subject to the equality constraint of volume.
Any suggestions?
Thank you.
 
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  • #2
In your TC function, remember that the box has four sides. Solve the volume constraint for, say, y. Then substitute that into TC. Now total cost is a function of only x. Minimize it. You certainly don't need lagrange multipliers.
 
  • #3
Thank you, I think this works.
 

1. What is the concept of minimizing total cost?

The concept of minimizing total cost is to find the most efficient and cost-effective way to produce a product or complete a task. This involves analyzing and reducing all the costs associated with the process, including materials, labor, and overhead expenses.

2. Why is it important to minimize total cost in manufacturing?

Minimizing total cost is important in manufacturing because it directly affects the profitability of the company. By reducing costs, companies can increase their profit margins, remain competitive in the market, and potentially lower the prices of their products for consumers.

3. How does minimizing total cost apply to making open boxes with squared bases?

In making open boxes with squared bases, minimizing total cost involves finding the most cost-effective way to produce the boxes while maintaining the desired quality and functionality. This includes optimizing the dimensions of the box, selecting the most affordable materials, and streamlining the production process.

4. What factors should be considered when trying to minimize total cost in making open boxes with squared bases?

When trying to minimize total cost in making open boxes with squared bases, factors such as the cost of materials, labor, transportation, and storage should be considered. Additionally, the efficiency of the production process, potential waste or defects, and the desired quality of the final product should also be taken into account.

5. How can technology be used to help minimize total cost in manufacturing open boxes with squared bases?

Technology can be used in several ways to help minimize total cost in manufacturing open boxes with squared bases. For example, computer-aided design (CAD) software can be used to optimize the dimensions and design of the box for maximum efficiency. Automation and robotics can also be implemented to streamline the production process and reduce labor costs. Additionally, data analytics can be used to identify areas for improvement and cost-saving opportunities in the manufacturing process.

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