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November 3rd, 2010, 05:05 PM   #1
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Optimization problem

a Donor is willing to build a new hangar for the program with the following stipulations
- the hangar must be in the shape of a half cylinder
- the hangar to have an exact volume of 225000 cubit feet
We would like to minimize the cost of the building. Currently, the construction costs for the foundation are $30 per square foot, the sides cost $20 per square foot to construct, and the roofing costs $15 per square foot. what should the dimensions of the building be to minimize the total cost ?

How can i solve for this !! help Thanks
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November 3rd, 2010, 05:20 PM   #2
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Re: Optimization problem

To simplify your question, I downloaded a pic

[attachment=0:uaud2u5l]???.GIF[/attachment:uaud2u5l]

Do you mean this?
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File Type: gif ???.GIF (23.2 KB, 1406 views)
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November 3rd, 2010, 06:40 PM   #3
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Re: Optimization problem

First, we should define the cost function. Let W be the width of the building, which will be the diameter of the half-cylinder, and let L be the length of the building, which will be the height of the half-cylinder. The area F of the floor will be F = L?W, the area R of the roof will be and the area S of the sides will be . The volume V of the building is .

Let:
be the cost per square foot for the foundation.
be the cost per square foot for the sides.
be the cost per square foot for the roof.

So the cost C, in dollars will be:



From the formula for volume, we see that , giving



Simplification yields:



Now, differentiating C with respect to W and equating to zero gives:



Multiply through by



Solve for W:





All that's left to do now is plug in the values given.
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November 3rd, 2010, 07:10 PM   #4
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Re: Optimization problem

Quote:
Originally Posted by genie
a Donor is willing to build a new hangar for the program with the following stipulations
- the hangar must be in the shape of a half cylinder
- the hangar to have an exact volume of 225000 cubit feet
We would like to minimize the cost of the building. Currently, the construction costs for the foundation are $30 per square foot, the sides cost $20 per square foot to construct, and the roofing costs $15 per square foot. what should the dimensions of the building be to minimize the total cost ?

How can i solve for this !! help Thanks
So, let be the width of the foundation, and be the length. If the hangar in the shape of half a cylinder, the max height of the vault should be.

The areas:

vault: ;

sidewall(both two): ;

foundation: .

The volume:


total cost: (let be the vaultcost, be the sidewallcost, be the foundationcost, where , , )



Construction an auxiliary function using (i)and (ii) as below:

.

Get the first order partial derivative of with .

;

;

Let (iv)and (v) equal to zeros, and with, we get a group of equations:





The solution of ,and







where , , , and
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November 3rd, 2010, 07:34 PM   #5
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Re: Optimization problem

I considered using optimization with constraint (Lagrange multipliers), but in the end decided to go with what I knew better.

Glad to see our answers agree.
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November 3rd, 2010, 07:38 PM   #6
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Re: Optimization problem

Quote:
Originally Posted by MarkFL
I considered using optimization with constraint (Lagrange multipliers), but in the end decided to go with what I knew better.

Glad to see our answers agree.


I checked my answer for a few times, and found blunders out. Fortunately, it follows what you got
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November 3rd, 2010, 08:13 PM   #7
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Re: Optimization problem

So the width should be 99.22 feet, the length should be 58.20 feet, the top height should be 49.61 feet,which will lead to the min cost...

And

Maybe the door should be in the roof not in the sidewall
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November 3rd, 2010, 08:33 PM   #8
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Re: Optimization problem

This results in a total cost (to the nearest penny) of $463,938.62
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November 3rd, 2010, 08:38 PM   #9
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Re: Optimization problem

Quote:
Originally Posted by MarkFL
This results in a total cost (to the nearest penny) of $463,938.62
Well well well, this's really a big money.
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November 3rd, 2010, 09:25 PM   #10
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Re: Optimization problem

Quote:
Originally Posted by stainburg
To simplify your question, I downloaded a pic

[attachment=0:rx0c59eq]???.GIF[/attachment:rx0c59eq]

Do you mean this?
yes, the picture you gave is such a hangar ( did u get it from a game or something ? lol ^^!
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