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I need help with this question:
The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log with diameter d?

What am I supposed to do?

Thx in advance_

  • Calculus -

    Assume the log to have a perfect circular section, of diameter d.
    The radius is therefore r=d/2.

    We have a choice of cutting a beam out of the log of width w, and height h, as long as sqrt(w²+h²)≤d.
    We can eliminate "h" at the source using equality and the above Pythagoras relation, i.e.

    Let the strength of the resulting rectangular beam be

    where k is a constant of proportionality.

    We look for the maximum value of S(w) by varying w, so we set dS/dw=0:
    Equating dS/dw=0 and solving for w:
    and therefore

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