What are the dimensions of the cedar chest that minimize the cost?

In summary, the problem is to find the dimensions of a cedar chest with a volume of 1440 dm^3 so that the cost is at a minimum. The cost of the lid is four times the cost of the rest of the chest, and the length is twice the width. To solve this, a function for the cost of the chest in terms of one variable must be found and differentiated. One approach is to assume the cost per dm^2 and use that to get a function in terms of one variable.
  • #1
Majestic_
3
0

Homework Statement


The length of a cedar chest is twice its width. The cost/dm^2 of the lid is four times the cost/dm^2 of the rest of the cedar chest. If the volume of the cedar chest is 1440 dm^3, find the dimensions so that the cost is a minimum.

Homework Equations


LWH = 1440
W = 2L

The Attempt at a Solution


I don't even know where to start. Can anyone help me get started? I've never seen a question like this in my examples, so I'm kind of lost.
 
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  • #2
Majestic_ said:

Homework Statement


The length of a cedar chest is twice its width. The cost/dm^2 of the lid is four times the cost/dm^2 of the rest of the cedar chest. If the volume of the cedar chest is 1440 dm^3, find the dimensions so that the cost is a minimum.

Homework Equations


LWH = 1440
W = 2L

The Attempt at a Solution


I don't even know where to start. Can anyone help me get started? I've never seen a question like this in my examples, so I'm kind of lost.

So, you have enough information to be able to write a function for the cost of this trunk. Now, as the various dimensions change, the cost is going to change. You want to find the point at which the cost of the trunk is at a minimum. So, start by getting a function for the cost of the trunk in terms of just one variable. Then, report back if you have more questions.
 
  • #3
Robert1986 said:
So, you have enough information to be able to write a function for the cost of this trunk. Now, as the various dimensions change, the cost is going to change. You want to find the point at which the cost of the trunk is at a minimum. So, start by getting a function for the cost of the trunk in terms of just one variable. Then, report back if you have more questions.

I already knew this. I just don't know where to get started to finding the function that needs to be differentiated (I've never solved a question like this before nor seen one in any of my examples).
 
  • #4
Oh, so is it finding the cost function that is giving you troubles?

I would start by assuming that the sides and the bottom of the trunk cost $1/dm^2, and that the lid costs $4/dm^2. Then, for example, the cost of the lid is $4 * l*w = $4 * 2w^2. So, you are going to want to get a function in w, then differentiate that one.
 

What is an optimization problem?

An optimization problem is a type of mathematical problem that seeks to find the best possible solution for a given situation. It involves maximizing or minimizing a certain objective function while considering a set of constraints.

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Some common types of optimization problems include linear programming, quadratic programming, integer programming, nonlinear programming, and dynamic programming.

What is the difference between global and local optimization?

Global optimization seeks to find the absolute best solution for a given problem, while local optimization focuses on finding the best solution within a specific region or range of values.

What are some common techniques for solving optimization problems?

Some common techniques for solving optimization problems include gradient descent, genetic algorithms, simulated annealing, and heuristic methods.

How are optimization problems used in real-world applications?

Optimization problems are used in a variety of real-world applications, such as in engineering, finance, logistics, and machine learning. They can help improve efficiency, reduce costs, and find the most optimal solutions to complex problems.

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