# Find Dimensions of a Log

• Feb 5th 2013, 11:38 PM
BobBali
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:

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

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

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

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...
• Feb 5th 2013, 11:45 PM
Prove It
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 \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 \displaystyle \begin{align*} 20\sqrt{2} \times 20\sqrt{2} \end{align*}.
• Feb 6th 2013, 12:05 AM
BobBali
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
• Feb 6th 2013, 12:24 AM
Prove It
Re: Find Dimensions of a Log
Then the answer you have been given is wrong.
• Feb 6th 2013, 05:56 AM
Soroban
Re: Find Dimensions of a Log
Hello, BobBali!

Quote:

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.