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Math Help - Maxima and Minima Problems

  1. #1
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    Maxima and Minima Problems

    The strenghth of a wooden beam with a rectangular cross section and fixed length is proportional to the product of its width and the square of its height where the proportionailty constant is 2 psi.



    Suppose the roofing company wants to make wooden beams of fixed length with rectangular cross sections from logs that are 4 inches in diameter. If the height of each beam must be at least 3.5 inches find the dimesnsions of the strongest beams.
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  2. #2
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    I honestly have no clue how to do this problem


    this is a maximum problem right?

    i dont even know where to start, the wording is confusing
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    bump
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  4. #4
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    really need this one guys
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  5. #5
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    Quote Originally Posted by pedromartinez View Post
    The strenghth of a wooden beam with a rectangular cross section and fixed length is proportional to the product of its width and the square of its height where the proportionailty constant is 2 psi.



    Suppose the roofing company wants to make wooden beams of fixed length with rectangular cross sections from logs that are 4 inches in diameter. If the height of each beam must be at least 3.5 inches find the dimesnsions of the strongest beams.
    Start by drawing a picture. You can take the cross section of a log to be the circle around the origin with radius 2: x^2+ y^2= 4. The rectangular cross section of a beam made from that log will be a rectangle with vertices at (x,y), (-x,y), (-x,-y), and (x, -y) where x and y satisfy x^2+y^2= 4. Now, if "w" is the width of the rectangle, x= w/2 so y= \sqrt{4- w^2/4} and the height of the rectangle is h= 2y= 2\sqrt{4- w^2/4}= \sqrt{16- w^2}.

    Since "The strenghth of a wooden beam with a rectangular cross section and fixed length is proportional to the product of its width and the square of its height where the proportionailty constant is 2 psi", strength= 2h^2w= 2(16- w^2)w= 32w- 2w^3.

    Differentiate that with respect to w to find the value of w that maximizes that.
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