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Math Help - Find Dimensions of a Log

  1. #1
    Junior Member BobBali's Avatar
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    Find Dimensions of a Log

    Good day to All,

    The question asked what are the dimensions of a square that can be cut from a cross-section of a tree log. The diameter of the log is given as 40cm. In my pic below it may not look like a square, but it is; sorry for the error produced in drawing it.

    So, what i first deduced that if x cm is taken out, we are left with a square of dimensions (40-2x).
    Then i found the area of the log's cross-section using:

    A= \pi (r)^2. This is = 1257cm^2

    Then I used length x width = Area to make a quadratic equation to solve for x:

    (-2x+40)(-2x+40) = 1257
    4x^2 - 160x +1600=1257
    4x^2 -160x +343= 0

    Solving the Quad equ gives: x = 37.4 or 2.3 (37.4 exceeds)
    The logical ans is 2.3, So> 40 - 2(2.3)= 36.4 cm
    However, the ans is 33.9cm...
    Thanks in advance.
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  2. #2
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    Re: Find Dimensions of a Log

    If the log is perfectly cylindrical, then the largest square will have a diagonal that's the same length as the diameter of your circular cross section.

    Since it's a square, the side lengths will be the same (x). Then we have by Pythagoras

    \displaystyle \begin{align*} x^2 + x^2 &= 40^2 \\ 2x^2 &= 1600 \\ x^2 &= 800 \\ x &= \sqrt{800} \\ x &= 20\sqrt{2} \end{align*}

    So the dimensions of the largest square will be \displaystyle \begin{align*} 20\sqrt{2} \times 20\sqrt{2} \end{align*}.
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  3. #3
    Junior Member BobBali's Avatar
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    Re: Find Dimensions of a Log

    This exactly what I did in the beginning and used the diagonal of 40cm to make a right-angled triangle...but this gives x= 28.3cm not 33.9cm
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    Re: Find Dimensions of a Log

    Then the answer you have been given is wrong.
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    Re: Find Dimensions of a Log

    Hello, BobBali!

    This exactly what I did in the beginning
    and used the diagonal of 40cm to make a right-angled triangle.
    But this gives x= 28.3cm, not 33.9cm

    Is there a typo?

    Their answer works if the diameter is 48 cm.
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