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The stiffness S of a rectangular beam is proportional to its width (w) times the cube of its depth/height (h). find the dimensions (i.e.: w and h) of stiffest beam that can be cut from a log which has a circular cross-section with diameter of 12-in.

  • math -

    You would use the relation of the circular cross-section of (w/2)²+(h/2)²=(12/2)² to relate h and w.
    After multiplying by 4 on both sides and rearranging, we have w=√(12²-h²).

    Now the stiffness has been defined as
    S=k(w)(h³) where k is a constant.

    If you substitute w from above then S is now a function of h only.

    Can you take it from here?

  • math -

    h^3=s/kw
    h^3=s/k(12-h)

    sorry, I'm still confused how to get the function of h. please help me to complete the assignment. thank you

  • math -

    You need to use the relation
    w=√12²-h²) to eliminate w from the function S.

    S(h)=k w h³
    =k √12²-h²) h³

    To find the maximum/minimum stiffness, you will proceed normally to equate
    S'(h) = 0
    to solve for h.

  • math -

    yes i understand now, thanks for your help

  • math :) -

    You're welcome!

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