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The strength of a rectangular beam is proportional to width*depth^2. What are the dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log?

  • calculus -

    let the width be w, and the depth be d
    S = kwd^2 where k is a constant
    but d^2 + w^2 = 144
    d^2 = 144-w^2

    S = kw(144-w^2)
    = 144kw - kw^3
    d(S)/dw = 144k - 3kw^2
    = 0 for max of S
    3kw^2 = 144k
    w^2 = 48
    w = √48
    then d = .....

    I will let you finish up.

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