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Math Help - Rectangular Wooden Beam

  1. #1
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    Rectangular Wooden Beam

    The strength of a rectangular wooden beam is proportional to the product of the width and the cube if its depth. If the beam is to be cut from a log in the shape of a cylinder of radius 3 feet, express the strength S of the beam as a function of the width x. Also, what is the domain of S?
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  2. #2
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    Quote Originally Posted by symmetry View Post
    The strength of a rectangular wooden beam is proportional to the product of the width and the cube if its depth. If the beam is to be cut from a log in the shape of a cylinder of radius 3 feet, express the strength S of the beam as a function of the width x. Also, what is the domain of S?
    Hello,

    I've attached a rough sketch of the situation.

    According to the problem

     S=x \cdot d^3

    From the diagram you know:

    x^2+d^2=(2r)^2 \Longleftrightarrow d=\sqrt{4r^2-x^2}

    Now plug in the term for d into the equation with S:

    S(x)=x \cdot \left(\sqrt{4r^2-x^2}\right)^3.

    To get real results it is necessary that the radicand is equal or greater than zero. Thus the value of x must be greater than zero and smaller than 2r:

    domain=\text{]}0, 2r\text{[}_{\mathbb R}

    EB
    Attached Thumbnails Attached Thumbnails Rectangular Wooden Beam-baumausstamm.gif  
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  3. #3
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    ok

    Great reply as always but how do you make such diagrams?
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  4. #4
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    earboth

    You said:

    "Now plug in the term for d into the equation with S."

    What exactly do you mean by this statement?
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  5. #5
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    Quote Originally Posted by symmetry View Post
    You said:

    "Now plug in the term for d into the equation with S."

    What exactly do you mean by this statement?
    Hello,

    from my previous post:

    d=\underbrace{\sqrt{4r^2-x^2}}_{\text{this is a term for d}}

    Now I took the RHS of this equation which contains the term for d and put it into the equation for S:

    S=x \cdot d^3

     S=x \cdot \left(\underbrace{\sqrt{4r^2-x^2}}_{\text{this is a term for d}} \right)^3


    EB
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  6. #6
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    ok

    I finally get it.

    Thanks!
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