Minimize Heating and Cooling Costs w/ Partial Derivatives

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Discussion Overview

The discussion revolves around a problem involving the optimization of heating and cooling costs for a building with a fixed volume, using partial derivatives and Lagrange multipliers. Participants are exploring the setup of the cost function and constraints related to the dimensions of the building.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a problem requiring the use of partial derivatives to minimize heating and cooling costs for a building with a volume constraint.
  • Another participant questions the necessity of using Lagrange multipliers, suggesting that the problem can be approached using only partial derivatives.
  • A participant emphasizes that the problem must incorporate constraints, indicating that Lagrange multipliers are essential for such cases.
  • There is a suggestion to express one variable as a function of others based on the volume constraint and to substitute it into the cost function.
  • Participants discuss the formulation of the cost function, with one proposing specific terms for the costs associated with different surfaces of the building.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the necessity of Lagrange multipliers versus using partial derivatives alone. Some participants argue for the elegance of Lagrange multipliers, while others believe the problem can be solved without them. The discussion remains unresolved on the preferred method of approach.

Contextual Notes

There are limitations in the clarity of the cost function formulation and the specific expressions for the surfaces involved. The discussion includes uncertainty about the correct setup of the optimization problem.

GoKings
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Hey,

This problem i need to use partial derivatives to solve but not Lagrange mulitpliers. My main problem is just setting it up:

A building in the shape of a rectangular box is to have a volume of 12,000 cubic feet. It is estimated that the annual heating and cooling costs will be $2 per square foot for the top,$4 per square for the front and back and $3 per square for the sides.(There is no cost for the bottom). Find the dimensions of the building that will result in a minimal annual heating and cooling cost. What is the minimum annual heating and cooling cost? Apply the second partial test to prove it is a minimum.
 
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What do you know about Lagrange multipliers...?

Daniel.
 
I know about langrage mulitplier but I have to solve this problem using both Langrane multipliers and using the normal method of partial derivatives
 
U can't.U have a constraint,which means that u have to use a Lagrange multipier.

Compute the cost function.

Daniel.
 
Well the first part of the questions ask to use partial derivatives and incorporating the constraint into the function to be minimized. While the second part asks to use Lagrange mulitpliers.
 
I don't know,it's pretty weird.Okay,i agree,it's much more elegant using Lagrange multipliers.But anyway,it can be done both ways and then u can compare & agree with me.

Okay.Express "z" as a function of "x" and "y" from the constraint and plug it in the cost function.That's the "ugly" way.In the end,to prove it's a minimum,u have to compute the hessian.

Daniel.
 
Ok but I am just confused on setting it up:
Is the constratint xyz=12,000
and the function=4x+3Y+2Z
?
 
It should involve products.The cost function,i mean.For example,the top should be 2xy.The sides should be 3zy for each of them and the front the remaining 4xz for each.

Add them & write the cost function...Then express "z" from the constraint as a function of "x" & "y"...

Daniel.
 

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