+0

# regions

0
55
1

0 lines cuts the plane into at most 1 region.
1 line cuts the plane into at most 2 regions.
2 lines cut the plane into at most 4 regions.

What is the most number of regions that 9 lines can cut the plane into?

Nov 18, 2019

#1
+106515
+1

Lines    Regions

1           2

2           4

3           7

4          11

Using just the regions we have and the sum of differences we have

2      4        7            11

2        3         4

1         1

We have   2 non-zero rows ....so   we will have a second power polynomial in the form

an^2 + bn  + c      and we have these system of equations

a  +  b  +  c   = 2

4a  + 2b + c = 4

9a  + 3b + c  = 7

Subtract  the first equation from each of the other two and we have the system

3a + b = 2     ⇒  -6a - 2b  = -4    (a)

8a + 2b  = 5          (b)

Add (a) and (b)  and we have

2a  = 1

a = 1/2

And

8(1/2) + 2b =5

4 + 2b =5

2b =1

b =1/2

And (1/2) + (1/2) + c = 2

1 + c  = 2

c =1

So....the  resulting polynomial for n lines  is

(1/2)n^2 + (1/2)n  + 1  =

[ n^2 + n  + 2 ]

___________

2

So....for 9 lines we have

9^2 + 9 + 2              92

_________  =       ____  =    46 regions

2                      2

Nov 18, 2019