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Math Help - Optimization

  1. #1
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    Optimization

    We just started this new section in optimization, and I don't really have a clue how to set up the word problems. I get that whatever one is attempting to minimize/maximize is what one takes the derivative of; and I also know that the expression that one takes the derivative of will have to be in terms of one variable (so the constraint equation would be used to do this) but I don't really understand how to set the expressions up... Can anyone help with the set up of the expressions for this problem?


    You have been hired as a consultant by a farmer who needs to build a grain silo. The silo is to have the shape of a cylinder, with a roof in the shape of a half-sphere (There is no bottom to the silo, as it is built on a platform). The silo must have a total volume of 2400pi cubic feet. Your job is to inform the farmer of the dimensions of the silo that will minimize the total cost of materials. The material for the side (the cylindrical part) costs $1.50 per sq ft, and the material for the roof (the half-sphere) costs $5 per sq ft.

    a) give the set-up and equations used to solve the problem
    b) give the values of r and h that minimize the cost
    c) evaluate for the total cost for the materials

    *My professor said something about giving the expression for the total surface area?

    I don't get how if you're given the total volume (2400pi cubic ft), you would write the equation in terms of surface area?
    ----> so, 2400pi = (4*pi*r^2)/2 + 2*pi*r*h (4*pi*r^2 is divided by 2 because the roof is only a half of a sphere)
    ----> then, divide the right side by pi?


    Obviously....I have no idea where to start?
    Any help would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by obsmith08 View Post
    You have been hired as a consultant by a farmer who needs to build a grain silo. The silo is to have the shape of a cylinder, with a roof in the shape of a half-sphere (There is no bottom to the silo, as it is built on a platform). The silo must have a total volume of 2400pi cubic feet. Your job is to inform the farmer of the dimensions of the silo that will minimize the total cost of materials. The material for the side (the cylindrical part) costs $1.50 per sq ft, and the material for the roof (the half-sphere) costs $5 per sq ft.

    a) give the set-up and equations used to solve the problem
    b) give the values of r and h that minimize the cost
    c) evaluate for the total cost for the materials

    V = \pi r^2 h + \frac{2}{3}\pi r^3 = 2400 \pi

    solve for h in terms of r ...

    h = \frac{2400}{r^2} - \frac{2r}{3}


    A = 2\pi r h + 2\pi r^2

    cost ...

    C = 3 \pi r h + 10\pi r^2

    C = 3\pi r \left(\frac{2400}{r^2} - \frac{2r}{3}\right) + 10\pi r^2

    C = \pi\left(\frac{7200}{r} + 8r^2\right)

    find \frac{dC}{dr} and determine the value of r that minimizes the cost.
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  3. #3
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    Smile

    you are a life saver! Thanks!
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