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calculus

Hi, I'm having trouble with this problem...

"If the strength of a rectangular beam of wood varies as its breadth and the square of its depth, find the dimensions of the strongest beam that can be cut out of a round log, diameter d. "

I can't come up with an equation with only 1 variable.

Help me, Please!!!
Many thanks, Stu.

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Creating a formula from 2 variables

  1. Im not really sure but i think its

    360(d)(120)

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    levy

  2. Hi Levy, any chance you could explain how you came up with that solution?
    I believe I should find values for Breath and Depth with relation to diameter, but I am stuck for a way to do it.

    Many thanks.

    1. 👍
    2. 👎
    Creating a formula from 2 variables

  4. rectangle must fit in circle of diameter d
    B =chord1 of circle = d sin (theta1/2)
    H = chord 2 of circle = d sin( [pi-theta]/2)
    because the angles subtended by the inscribed breadth and depth add up to a straight line. pi is straight line angle in radians.
    S = k B H^2
    S = k d sin (theta/2)*d^2 sin^2 ([pi-theta]/2)
    let A = theta/2 to make it all easier
    S = k d^3 sin A sin^2 (pi/2-A)
    but sin (pi/20-A) = cos(A)
    so
    S = k d^3 sin(A) cos^2(A)
    take derivative and set to 0
    0 = dS/dA = k d^3[ -2 sin^2(A)cos(A)+ cos^3(A)]
    2 sin^2 (A) = cos^2(A)
    but cos^2(A) = 1 - sin^2(A)
    2 sin^2 A = 1 - sin^2(A)
    3 sin^2 (A) = 1
    sin(A) = sqrt(3)/3 = about .577
    so A is about 35.2 degrees
    so
    B = d sin (A) = .577 d
    H = d sin(90-A) =d sin 54.8deg = .817 d

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    2. 👎
    Damon

  5. Damon, you're a God-send!

    If you're ever in North Wales I'll buy you a pint!

    1x10^3 Thanks.

    Stu.

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    Creating a formula from 2 variables

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