November 3rd, 2010, 05:05 PM  #1 
Newbie Joined: Nov 2010 From: CA Posts: 25 Thanks: 0  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 
November 3rd, 2010, 05:20 PM  #2 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Optimization problem
To simplify your question, I downloaded a pic [attachment=0:uaud2u5l]???.GIF[/attachment:uaud2u5l] Do you mean this? 
November 3rd, 2010, 06:40 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  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 halfcylinder, and let L be the length of the building, which will be the height of the halfcylinder. 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. 
November 3rd, 2010, 07:10 PM  #4  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Optimization problem Quote:
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  
November 3rd, 2010, 07:34 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  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. 
November 3rd, 2010, 07:38 PM  #6  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Optimization problem Quote:
I checked my answer for a few times, and found blunders out. Fortunately, it follows what you got  
November 3rd, 2010, 08:13 PM  #7 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  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 
November 3rd, 2010, 08:33 PM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  Re: Optimization problem
This results in a total cost (to the nearest penny) of $463,938.62 
November 3rd, 2010, 08:38 PM  #9  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Optimization problem Quote:
 
November 3rd, 2010, 09:25 PM  #10  
Newbie Joined: Nov 2010 From: CA Posts: 25 Thanks: 0  Re: Optimization problem Quote:
 

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