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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

dierdurst Aug 23, 2018

#1**+2 **

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

CPhill Aug 24, 2018