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The field shown has been planted uniformly with wheat. At harvest, the wheat at any point in the field is brought to the nearest point on the field’s perimeter. What is the fraction of the crop that is brought to the longest side?



- Daisy

 Aug 23, 2018
edited by dierdurst  Aug 23, 2018

I'm not sure....but here's my best effort



Let ABCD  be the  trapezoidal field


Its  area   = 

(1/2)(height) (DC + AB) = 

(1/2)(50√3) (100 + 200) =

7500√3   m^2  


And  the  crop will be brought to the longer side whenever it is contained within  the trapezoid  AEFB


Reasoning :

DB bisects  angle CBA....angle FIB  = FKB  and FB  = FB

So....by AAS, triangle FIB  = triangle FKB

So....any crop harvested within triangle FIB will be closer to IB than BK


For the same reason any crop harvested within triangle EJA  will be closer to JA  than side AL

And any crop harvested within EFIJ  will be closer to the longest side than to side DC


And the area of AEFB  =

(1/2)(EJ) (AB + EF)  =

(1/2)(25√3) (200 + 50) =

3125√3  m^2


So  the fraction of the crop brought to the longest side  is


3125√3  / 7500√3  =   5/12



cool cool cool

 Aug 24, 2018

Thank you!


- Daisy

dierdurst  Aug 24, 2018

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