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Working out a formula for 2 variables?

  1. Jan 29, 2008 #1
    Hi Folks,

    I'm working through a pre-selection maths course and have come to a complete halt with the following Differentiation 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. "

    It's the only problem in the notes that has more than 1 variable and no example to show how its done.

    Here's what I've come up with so far....
    S varies with Breadth, therefore S=kB
    S varies with Depth^2, therefore S=kD^2
    The derivative of a constant term = 0

    If I could work out the formula I think I'd be ok, but I'm completely at a loss for what to do next.

    Thanks for any help you can give me, and sorry it's a little basic.

  2. jcsd
  3. Jan 29, 2008 #2


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    Hint: Can you related B and D to d?
  4. Jan 29, 2008 #3
    Hi Hootenanny,

    I think I tried this last night, I came up with

    therefore S=(d-2x)(d-2y)(d-2y)

    Still with 2 variables. I'm assuming x and y cannot be equal as the problems states a rectangular beam.

    Am I along the right lines or 90 degrees to them?
  5. Jan 29, 2008 #4


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    Your on the right lines. Draw yourself a semi-circle, mark on your diameter together with B and 1/2D as chords. Now, join the two endpoints of B and 1/2D with a further chord. Now, what is the maximum size of this chord?
    Last edited: Jan 29, 2008
  6. Jan 29, 2008 #5
    I'm really, Really sorry Hootenanny, I was right with you up until " together with 1/2B and 1/2D as chords. " Where on the semi-circle should I be marking the chords? Do you mean 1/2 Breadth or 1/2 the distance from the edge of the beam to the circle?

    I'm sorry to seem so dumb, but I really am dumb.
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