Constrained Minimization Problem(HELP )

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In summary, a rectangular box with dimensions of x, y, and z should be created with heavy-duty cardboard costing $0.36 per square foot for the top and bottom, and lightweight cardboard costing $0.06 per square foot for the sides. The cost of the box can be minimized by using the equation A=2xy+2xz+2yz for surface area and V=xyz=162 for volume, and creating a cost function by multiplying the area of each type of cardboard by its respective cost per area and adding them together.
  • #1
MathNoob123
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Homework Statement


A closed rectangular box is made with two kinds of materials. The top and bottom are made with heavy-duty cardboard costing $0.36 per square foot, and the sides are made with lightweight cardboard costing $0.06 per square foot. Given that the box is to have a capacity of 162 cubic feet, what should its dimensions be if the cost is to be minimized?


Homework Equations


Surface Area A=2xy+2xz+2yz
Volume= V=xyz=162

partial derivitations are required to do this(which i know how to do)


The Attempt at a Solution


The hard part that I am trying to figure out is creating the Cost equation. All I need to know is what that equation would be and I can take care of everything afterwards. Like mentioned before in (2), the surface area A=2xy+2xz+2yz, but I have no idea how I am going to create a cost function including the prices mentioned in the problem. PLEASE HELP. BLESS ANYONE WHO DOES. THANK YOU SO MUCH IN ADVANCE!
 
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  • #2
MathNoob123 said:
… The top and bottom are made with heavy-duty cardboard costing $0.36 per square foot, and the sides are made with lightweight cardboard costing $0.06 per square foot.
…
I have no idea how I am going to create a cost function including the prices mentioned in the problem.

Hi MathNoob123! :wink:

Multiply the area of heavy-duty cardboard by the cost per area, and multiply the area of lightweigh cardboard by the cost per area, and add :smile:
 
  • #3
tiny-tim said:
Hi MathNoob123! :wink:

Multiply the area of heavy-duty cardboard by the cost per area, and multiply the area of lightweigh cardboard by the cost per area, and add :smile:


I have already solved the problem, but thank you very much for replying. Really appreciate it.
 

1. What is a constrained minimization problem?

A constrained minimization problem is a mathematical optimization problem where the goal is to find the minimum value of a function, subject to a set of constraints. The constraints limit the possible values that the variables can take on.

2. What types of constraints are commonly used in constrained minimization problems?

Commonly used constraints include equality constraints, where the variables must satisfy a set of equations, and inequality constraints, where the variables must satisfy a set of inequalities. Other types of constraints include bounds on the variables and linear or nonlinear constraints.

3. How is a constrained minimization problem solved?

There are various methods for solving constrained minimization problems, such as the Lagrange multiplier method, the penalty function method, and the interior-point method. These methods involve finding the critical points of the objective function, subject to the constraints.

4. What are some real-life applications of constrained minimization problems?

Constrained minimization problems have many practical applications in fields such as engineering, economics, and operations research. For example, they can be used to optimize the design of structures, minimize production costs, and maximize profits.

5. What are some challenges of solving constrained minimization problems?

One of the main challenges of solving constrained minimization problems is finding an efficient and accurate solution, as these problems can become increasingly complex as the number of variables and constraints increases. Additionally, selecting the appropriate method for a particular problem can also be a challenge.

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