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Math Help - calc optimization help

  1. #1
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    calc optimization help

    Alright! My final question on here till next semester lol. Doing some last minute review for my final tonight and came across this problem having a little trouble with it.

    A rectanglular storage container with an open top is to have a volume of 10 cubic-meters. Material for the sides costs $6 per square-meter. Material for the base costs $10 per square-meter. Find the cost of materials for the cheapest container.

    Thanks

    {edit} ah yes sorry. Length of its base is twice the width.
    Last edited by UMStudent; May 9th 2007 at 10:05 AM.
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  2. #2
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    I would say that you are missing a piece of information. Are you sure that you typed the entire problem?
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  3. #3
    Bar0n janvdl's Avatar
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    Quote Originally Posted by UMStudent View Post
    Alright! My final question on here till next semester lol. Doing some last minute review for my final tonight and came across this problem having a little trouble with it.

    A rectanglular storage container with an open top is to have a volume of 10 cubic-meters. Material for the sides costs $6 per square-meter. Material for the base costs $10 per square-meter. Find the cost of materials for the cheapest container.

    Thanks

    {edit} ah yes sorry. Length of its base is twice the width.
    L = 2x
    W = x
    H = h

    So Volume = 2x^2h = 10

    Exterior Area = 2(L + B)H + (L x B) [Because you do not have a lid]
    = 2(2x + x)h + (2x . x)
    = 4x^2h + 2x^2
    = 6xh + 2x^2

    But we know that h = (10)/(2x^2)

    = 6x((10)/(2x^2)) + 2x^2
    = 60x/2x^2 + 2x^2
    = 30/x + 2x^2

    dA/dx = -30/x^2 + 4x = 0
    -30 = -4x^3
    7,5 = x^3
    x = 1,957433821

    Now you can calculate the rest.

    This is your sides:30/x

    And this is your bottom lid:2x^2
    Last edited by janvdl; May 9th 2007 at 10:30 AM.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by UMStudent View Post
    Alright! My final question on here till next semester lol. Doing some last minute review for my final tonight and came across this problem having a little trouble with it.

    A rectanglular storage container with an open top is to have a volume of 10 cubic-meters. Material for the sides costs $6 per square-meter. Material for the base costs $10 per square-meter. Find the cost of materials for the cheapest container.

    Thanks

    {edit} ah yes sorry. Length of its base is twice the width.
    Let the width be w metres, then the length is 2w metres, and as the volume
    is 10 cubic metres the height is: 5/w^2 metres.

    Total area of the sides is:

    As = 2*w*5/w^2 + 4*w*5/w^2

    ... = 30/w

    The area of the base is:

    Ab = 2*w^2

    So the cost is:

    C = 180/w + 20*w^2

    Now you need to find the value of w that minimises C, and that minimum
    cost is your answer

    RonL
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  5. #5
    Bar0n janvdl's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Let the width be w metres, then the length is 2w metres, and as the volume
    is 10 cubic metres the height is: 5/w^2 metres.

    Total area of the sides is:

    As = 2*w*5/w^2 + 4*w*5/w^2

    ... = 30/w

    The area of the base is:

    Ab = 2*w^2

    So the cost is:

    C = 180/w + 20*w^2

    Now you need to find the value of w that minimises C, and that minimum
    cost is your answer

    RonL
    Is my solution wrong Captain?
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  6. #6
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    Ok
    let x, y, z be the dimensions of the box.
    Since volum should be 10, (1) xyz=10
    Length of its base is twice the width: (2) y=2z
    (3) Cost=6(2xy+2xz)+10yz=12xy+12xz+10yz
    You need a function cost having just one variable, so let's use (1) and (2) to simplify (3)
    (2) into (1) gives 2xz^2=10 or x=5/z^2
    Using this relation and (2) into (3) gives:
    cost=20z^2+180/z
    Solve the equation Cost'=0 to find z which gives minimum cost.
    40z-180/z^2=0
    40z^3-180=0
    z^3=4.5
    z=1.65
    x=1.84
    y=3.3
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  7. #7
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    Quote Originally Posted by janvdl View Post
    Is my solution wrong Captain?
    Your solution is wrong since you did not minimize the cost, but the area.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by janvdl View Post
    Is my solution wrong Captain?
    I havn't checked it, you posted while I was writting mine, so I did not see
    yours. If one is wrong its just as likely to be mine.

    RonL
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  9. #9
    Bar0n janvdl's Avatar
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    Oh of course! He pays diff prices for diff parts of the box.
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