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Math Help - Optimizing minimal cost

  1. #1
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    Optimizing minimal cost

    This problem is on optimizaton.
    An airplane hanger (looks like a right circular cylinder cut in half long ways) must have a volume of exactly 225,000 cubic feet. Want to minimize cost of building it when the foundation cost $30 per square foot, siding cost $20 per sq ft, and roofing cost $15 per sq ft. What are the dimensions of the hangar to achieve minimal cost???? Also, roofing costs fluctuate so what are the dimensions of the hangar if the cost of roofing is $R per sq ft?
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    Dear zackmo11,

    1st step: declare the cost function:
    cost = area_found*rate_found + surface_side*rate_side+surface_proof*rate_proof.

    2nd step: let r and h the radius and the height of the semi-cylinder.Express the area_found and the 2 kind of surface with r and h

    3th step: Look for a connection between h and r. We have to remark that r and h is not independent because volumen V is fix --> V(r, h) = const.

    4th step: express the cost function with only one variant (r or h). So we get cost(r) function.

    5th step: look for the minimum
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    Need more info please!!!

    Im looking for dimensions of the hangar if cost is $R (R is a variable) per sq foot and the dimensions for minimal cost. I'd like to see more than just words though, it didnt make enough sense for me to do the problem. I need some work to follow, with the numbers given.
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    Look at an example firstly with numbers secondly with parameter.

    We search the maximum of the function:
    f(x) = -2x*x + 3*x

    The method is femous:
    f' = 0
    -4*x + 3 = 0
    x = 3/4

    okey, now we seach the maximum of the function:
    f(x) = -2*x*x + rate*x, where rate is a fix parameter.

    The method is same:
    f' = 0
    -4*x + rate = 0
    4*x = rate
    x = rate/4
    This is the answer depending on the parameter.
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  5. #5
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    nice

    I like those words of wisdom friend. You've misinterpreted me though. For
    these optimization problems the hardest part is coming up with an equation for
    the word problem. Thats what i cant get, the rest is algrebra taking a derivative and setting equal to zero. The equation would be volume(225,000)= something. I'm not sure how to use the costs given -in the equation. So I'm stuck on the very beginning.
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  6. #6
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    Quote Originally Posted by zackmo11 View Post
    This problem is on optimizaton.
    An airplane hanger (looks like a right circular cylinder cut in half long ways) must have a volume of exactly 225,000 cubic feet. Want to minimize cost of building it when the foundation cost $30 per square foot, siding cost $20 per sq ft, and roofing cost $15 per sq ft. What are the dimensions of the hangar to achieve minimal cost???? Also, roofing costs fluctuate so what are the dimensions of the hangar if the cost of roofing is $R per sq ft?
    1. Let the hangar have a length x and a height (= radius of circle) y. Draw a diagram.

    Then you have:

    \text{Cost} = (2yx)(30) + (\pi y^2) (20) + (\pi yx)(15) = \, .... .... (1)

    (this is left for you to simplify)

    subject to the condition

    225,000 = \frac{\pi y^2 x}{2} .... (2)

    Use equation (2) to get x in terms of y. Substitute this expression for x into equation (1) to get cost as a function only of y. Use calculus to find the value of y that minimises cost. etc.

    -----------------------------------------------------------------------------------------

    2. \text{Cost} = (2yx)(30) + (\pi y^2) (20) + (\pi yx)(R) = \, ....

    Proceed in the same way as above. You'll get y in terms of R this time. As a check, substitute R = 15 (you should get the answer found above).
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  7. #7
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    what?

    Thats so much better, but i dont understand how you derived the cost= equation. is it based off the volume equation of a cylinder? And i solve the cost= equation for R right?
    Last edited by zackmo11; December 8th 2008 at 06:28 PM. Reason: question answered! mostly
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  8. #8
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    Quote Originally Posted by zackmo11 View Post
    Thats so much better, but i dont understand how you derived the cost= equation. is it based off the volume equation of a cylinder? And i solve the cost= equation for R right?
    Cost = (area of floor)(30) + (area of sides)(20) + (area of roof)(R).

    area of floor = area of rectangle.

    area of sides = area of two semi-circles = area of circle.

    area of roof = half the area of the curved surface of a cylinder.

    R = $15 per sq ft in part 1.
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