Minimize Heating and Cooling Costs w/ Partial Derivatives

In summary: Okay so the cost function should be (4x+3y+2z) and "z" from the constraint should be (x+y).Then to prove it's a minimum,u have to compute the hessian.
  • #1
GoKings
4
0
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.
 
Physics news on Phys.org
  • #2
What do you know about Lagrange multipliers...?

Daniel.
 
  • #3
I know about langrage mulitplier but I have to solve this problem using both Langrane multipliers and using the normal method of partial derivatives
 
  • #4
U can't.U have a constraint,which means that u have to use a Lagrange multipier.

Compute the cost function.

Daniel.
 
  • #5
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.
 
  • #6
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.
 
  • #7
Ok but I am just confused on setting it up:
Is the constratint xyz=12,000
and the function=4x+3Y+2Z
?
 
  • #8
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.
 

1. What are partial derivatives and how do they relate to minimizing heating and cooling costs?

Partial derivatives are a mathematical concept used to calculate the rate of change of a function with respect to one of its variables, while holding the other variables constant. In the context of minimizing heating and cooling costs, partial derivatives are used to determine how changes in different factors, such as temperature and insulation, affect the overall cost.

2. How can partial derivatives be used to optimize heating and cooling systems?

By taking partial derivatives of the cost function with respect to different variables, such as temperature and insulation, we can identify which factors have the greatest impact on the overall cost. This information can then be used to optimize heating and cooling systems, such as adjusting thermostat settings or improving insulation, to minimize costs.

3. Are there any limitations to using partial derivatives for minimizing heating and cooling costs?

Partial derivatives are a useful tool for optimizing heating and cooling costs, but they do have limitations. They assume that the cost function is continuous and differentiable, and they may not account for all the complexities of real-world heating and cooling systems. Other factors, such as weather patterns and user behavior, may also impact overall costs.

4. Can partial derivatives be used for both heating and cooling systems?

Yes, partial derivatives can be used for both heating and cooling systems. The same principles apply for both, but the specific variables and cost functions may differ. For heating systems, the cost function may be related to the amount of energy used, while for cooling systems, it may be related to the amount of energy saved.

5. How can partial derivatives help in making decisions about energy-efficient upgrades?

Partial derivatives can provide valuable insights into which factors have the greatest impact on heating and cooling costs. This information can be used to make informed decisions about energy-efficient upgrades, such as investing in better insulation or upgrading to a more efficient HVAC system, in order to minimize overall costs in the long term.

Similar threads

MHB Solve Optimization Problem: Minimize Cost of New Highway
    • Calculus
Replies
2
Views
4K
MHB Minimize the cost of the fence
    • Calculus
Replies
2
Views
3K
{Calculus} Minimize the heat loss
    • Calculus and Beyond Homework Help
Replies
9
Views
3K
HVAC Exhaust flow for the chimney design on a custom boiler
    • DIY Projects
Replies
8
Views
2K
MHB Solve Box Cost Minimization w/ Lagrange Multipliers
    • General Math
Replies
1
Views
2K
MHB Express material cost as function of radius
    • General Math
Replies
2
Views
7K
OPTIMIZATION: Minimizing Packaging Costs
    • Calculus and Beyond Homework Help
Replies
3
Views
6K
I minimizing the cost of this building? immediate help
    • Calculus and Beyond Homework Help
Replies
3
Views
3K
Optimizing Air Flow for CO2 and Heat Loss in Vans
    • Biology and Medical
Replies
7
Views
2K
Applied Optimization Help
    • Calculus and Beyond Homework Help
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top