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The simultaneous conditions x - y <= 6, x + y <= 6, and x >= 0 define a region R. What is the area of region R? How many lattice points are in region r? How many lattic points lie on the boundary of R ? What if 6 is replaced by 100?

 Jul 14, 2021
 #1
avatar+121065 
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See  the  graph  here  :  https://www.desmos.com/calculator/qzfidyzqfw

 

We  have  two  triangles  whose  base and  height   =  6....so...the  area =   6 * 6    = 36

 

By Pick's Theorem

 

Area =   Number of interior lattice points  +  Number  of lattice points on the  boundary/2   - 1

 

The  number of lattice points on the  boundary  =    1 + 6  + 6 +  11  =   24

 

So

 

36 =   I  +  24 /2    -  1

 

36  = I  +  12 -  1

 

36 = I  + 11

 

I =  25  =  25 interior lattice points

 

 

If  6 is replaced  by 100.....the  area =   100^2 =  10000

The  number of lattice points on the  boundary =  1 + 100 + 100 + 199 = 400

 

By Pick's Theorem

 

10000  =  I   +  400/2  - 1

 

10000 = I  + 200 - 1

 

10000 =  I  +  199

 

I  =   9801   lattice points in the  interior

 

EDIT  TO CORRECT A  PREVIOUS ERROR   !!!!

 

cool cool cool

 Jul 14, 2021
edited by CPhill  Jul 14, 2021
edited by CPhill  Jul 14, 2021
 #2
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wrong...

Guest Jul 14, 2021
 #3
avatar+121065 
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Oops!!!....see  my edit above   !!!!

 

 

 

cool cool cool

CPhill  Jul 14, 2021

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