La_matematica_degli_indovinelli_3.0

“La matematica degli indovinelli” di Matteo Puzzle – matematicare@hotmail.com
Considerazioni geometriche per il calcolo dei raggi:
⎛π
⎞ 3
r1 = C1H = ⋅ tan ( C1BH)
= ⋅ tan ⎜ ⎟=
2 2 ⎝ 6 ⎠ 6
2 3
( ) ( ⎛π
⎞ ⎛π
⎞ ⋅
3
r2 = C1S⋅tan K1C1S ⋅ tan C2K1S)
= r1⋅ tan ⎜ ⎟ = ⋅ tan ⎜ ⎟ =
⎝ 6 ⎠ 2 ⎝ 6 ⎠ 18
r
3
⋅
5
π ⎞ ⋅
3
⎟
⎠
7
π ⎞ ⋅
3
⎟
⎠
2 n 1
π ⋅ +
⎛
= ⋅ tan ⎜ =
2 ⎝ 6 54
⎛
r4
= ⋅ tan ⎜ =
2 ⎝ 6 162
............
⎛ ⎞
rn
= ⋅tan
⎜ ⎟ con n∈N
2 ⎝ 6 ⎠
calcolo dell’area degli infiniti cerchi:
2 1 2
∞ ∞ 2 2 ∞
2 ∞ 2 ⋅n
⋅ n+
−( 2⋅ n+
1
⎛ ⎞ ⎛
)
⎞
⎛π ⎞ ⋅3 π ⋅ ⎛ 1 ⎞ π ⋅
⎛1⎞
Λ
C
= π ⋅∑ ⋅ tan
= π ⋅ = ⋅ = ⋅
⎜ ⎜ ⎟ 2⋅ n+
1
n= 0 2 6 ⎟ ∑ ⎜ ⎟ ⎜ ⎟
⎝ ⎠
⎜
n= 0 4 ⎟ ∑ ∑
⎝
⎠ ⎝ ⎠ 4 n=
0⎝3 ⎠ 12 n=
0⎝3⎠
poiché (essendo una serie geometrica):
∞
ik
kn a
kn a r ⋅
⋅
∞
⋅ ⋅
∑a⋅ r = o più in generale: a r
k
∑ ⋅ =
k
n= 0 1−r
n=
i 1−r
⎧a
∈
⎪
con ⎨ − 1 < r < 1
⎪
⎩k
> 0
2
⎧ π ⋅
⎪a
=
⎪
12
⎪ 1
in questo caso ⎨r
=
⎪ 3
⎪ k = 2
⎪
⎩
quindi, l’area occupata dagli infiniti cerchi è:
2
3⋅π
⋅ Λ
C
= 0,2946⋅
32
2
area % occupata dagli infiniti cerchi rispetto all’intero triangolo equilatero:
2
3⋅π
⋅
100⋅
100⋅ΛC
32 25⋅π
⋅ 3
Λ
C
% = = = 68%
2
ΛT
⋅ 3 2
4
36