# Minimization Problem

1. Apr 1, 2005

### GoKings

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.

2. Apr 1, 2005

### dextercioby

What do you know about Lagrange multipliers...?

Daniel.

3. Apr 1, 2005

### GoKings

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. Apr 1, 2005

### dextercioby

U can't.U have a constraint,which means that u have to use a Lagrange multipier.

Compute the cost function.

Daniel.

5. Apr 1, 2005

### GoKings

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. Apr 1, 2005

### dextercioby

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. Apr 1, 2005

### GoKings

Ok but im just confused on setting it up:
Is the constratint xyz=12,000
and the function=4x+3Y+2Z
?????

8. Apr 1, 2005

### dextercioby

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.