SOLUTION: I was wondering if you could help me with this question! I dont even know where to start! Thank you! It is estimated that the average diameter of peeled logs coming into your sawm

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Question 106902: I was wondering if you could help me with this question! I dont even know where to start! Thank you!
It is estimated that the average diameter of peeled logs coming into your sawmill is 14 inches.
a) What is the thickness of the largest square timber that can be cut from the average log?
b)If the rest of the log is made into mulch, what percent of the original log is the square timber?
Answer by Fombitz(13823) About Me  (Show Source):
You can put this solution on YOUR website!
In other words, you have a 14" circle, what is the largest square that can fit inside of your circle.
drawing%28+300%2C+300%2C+-1.2%2C+1.2%2C+-1.2%2C+1.2%2Cgrid%28+1+%29%2Ccircle%28+0%2C+0%2C+1+%29%2C%0D%0A++red%28line%28+0%2C+0%2C+.707%2C+.707%29%29%2C%0D%0Alocate%28.1%2C.55%2C7+in%29%2C%0D%0Alocate%28.5%2C-.2%2Cs%29%2C%0D%0Agreen%28line%28+-.707%2C+.707%2C+.707%2C+.707%29%29%2C%0D%0Agreen%28line%28++.707%2C+.707%2C+.707%2C-.707%29%29%2C%0D%0Agreen%28line%28++.707%2C-.707%2C-.707%2C-.707%29%29%2C%0D%0Agreen%28line%28+-.707%2C-.707%2C-.707%2C+.707%29%29%29
The diagonal line is also the radius of the circle (7 inches).
The diagonal line is also the hypotenuse of the right, isoceles triangle with sides s,s, and 7 inches.
Since it's a right triangle, you can use the Pythagorean theorem.
s%5E2%2Bs%5E2=7%5E2
2s%5E2=7%5E2
s=7%2Fsqrt%282%29
The side of the square is 2s long, so the largest square that can be cut from the circle will be
t=2s=14%2Fsqrt%282%29
t=9.9 inches
b) The area of the circle equals the area of the square plus the leftover(mulch).
1.A%5Bc%5D=A%5Bs%5D%2BA%5Bm%5D
Percentage of mulch area to total area is equal to
A%5Bm%5D%2FA%5Bc%5D
Divide equation 1. by A%5Bc%5D
1.A%5Bc%5D%2FA%5Bc%5D=A%5Bs%5D%2FA%5Bc%5D%2BA%5Bm%5D%2FA%5Bc%5D
1=t%5E2%2F%28pi%29R%5E2%2BA%5Bm%5D%2FA%5Bc%5D
A%5Bm%5D%2FA%5Bc%5D=1-t%5E2%2F%28pi%29R%5E2
A%5Bm%5D%2FA%5Bc%5D=1-%2814%5E2%2F2%29%2F%28%28pi%29%2A7%5E2%29
A%5Bm%5D%2FA%5Bc%5D=1-%28128%29%2F%28153.9%29
A%5Bm%5D%2FA%5Bc%5D=.168
16.8% of the tree ends up as mulch.
83.2% of the tree ends up as the square.