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Math Help - [SOLVED] optimization problem (cost function)

  1. #1
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    [SOLVED] optimization problem (cost function)

    I can find the cost function, but I don't know how to find minimum cost. Which should = $329.34

    A rectangular storage container with an open top is to have a volume of 20 m^3. The length of its base is twice the width. Material for the base costs $15 per square meter.Material for the sides costs $7 per square meter.
    1. Find the cost function for the container.
    2. Find the cost of materials for the cheapest such container.

    <br />
V=20=2x^2y <br />
    <br />
Surface Area = 2x^2+6xy<br />
    Cost Function:
    <br />
15*(2x^2) + \frac{7*60}{x}<br />
    <br />
30x^2 + \frac{420}{x}<br />
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  2. #2
    Member SengNee's Avatar
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    Quote Originally Posted by coolguy00777 View Post
    I can find the cost function, but I don't know how to find minimum cost. Which should = $329.34

    A rectangular storage container with an open top is to have a volume of 20 m^3. The length of its base is twice the width. Material for the base costs $15 per square meter.Material for the sides costs $7 per square meter.
    1. Find the cost function for the container.
    2. Find the cost of materials for the cheapest such container.

    <br />
V=20=2x^2y <br />
    <br />
Surface Area = 2x^2+6xy<br />
    Cost Function:
    <br />
15*(2x^2) + \frac{7*60}{x}<br />
    <br />
30x^2 + \frac{420}{x}<br />

    To find the max or min,you just find the stationary point of the graph and identify its concavity.

    \frac{dy}{dx}=0
    \frac{d^2y}{dx^2} is positive or megative.
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  3. #3
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    sorry, but I'm still lost. Is there not an algebraic way to find the answer? What part do I take the derivative of?
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  4. #4
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    skeeter's Avatar
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    Quote Originally Posted by coolguy00777 View Post
    sorry, but I'm still lost. Is there not an algebraic way to find the answer? What part do I take the derivative of?
    your cost function ...

    \frac{d}{dx}\left[C = 30x^2 + \frac{420}{x}\right]

    \frac{dC}{dx} = 60x - \frac{420}{x^2} = 0

    x = \sqrt[3]{7}

    C(\sqrt[3]{7}) = 329.34
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