La_matematica_degli_indovinelli_3.0

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“La matematica degli indovinelli” di Matteo Puzzle – matematicare@hotmail.com area totale degli infinti cerchi inscritti nel quadrato: ∞ 2⋅n 2 ∞ 4⋅n ⎛ 2 2 2 3 2 2⎞ 2 ⋅ − Λ C = π ⋅ r + 4⋅π ⋅∑( r⋅( 2− 1) ) = π ⋅ r + 4⋅π ⋅∑( r ⋅( 2− 1) ) = π ⋅r ⋅⎜ ⎜ n= 1 n= 1 2 ⎟ ⎝ ⎠ ⎛3⋅ 2−2⎞ Λ C = π ⋅r ⋅ ⎜ 3,5227⋅r 2 ⎟ ⎝ ⎠ 2 2 area % occupata dagli infiniti cerchi rispetto all’intero quadrato: ⎛ 2 3⋅ 2−2⎞ 100⋅π ⋅r ⋅⎜ ⎟ 100⋅Λ 2 π ⋅ 2⋅( 3− 2) C Λ C % = = ⎝ ⎠ = 88% 2 Λ 2⋅ r 8 Q ( ) 39
“La matematica degli indovinelli” di Matteo Puzzle – matematicare@hotmail.com Indovinello 10 - “La convergenza dei cerchi in un triangolo isoscele” ACB = α AB = l’area occupata dagli infiniti cerchi che convergono verso il suddetto angolo α è: ∞ ⎛ ⎛π −α ⎞ Λ C = π ⋅ ⋅tan ⎜ ⎜ ⎟ n= 1 ⎝2 ⎝ 4 ⎠ con 0 < α < π 2⋅n−1 ( 2 Λ C = π ⋅ ⋅ f α ) ∑ indicando con f ( α ) la funzione d’angolo: f ( α ) ⎞ ⎟ ⎠ 2 ⎛ 3 ⎛ α 3 2 2 2 2 2 2 ⎞ ⎞ ⎜ ⎛ ⎞ 8⋅sin ( α) ⋅cos⎜ ⎟ ⋅( 3⋅ cos( α) + 9⋅ cos( α) + 23⋅ sin ( α) + 12) + ( cos( α) + 1) ⋅ cos ( α) ⋅ sin ( α) + 2 + 8⋅cos α ⋅ 1−4⋅ sin α + 3⋅ 31⋅ sin α + 2 ⎟ ⎜ ⎜ 2 ⎟ ⎝ ⎝ ⎠ ⎠ ⎟ = ⎜ − 4 3 4 ⎟ ⎜ 2 ⎛ ⎛α ⎞ 2 ⎛α ⎞ ⎛ ⎛α ⎞ 2 ⎞ ⎛ ⎛α ⎞ ⎞ ⎞ ⎟ 16⋅ ( cos( α) + 1) ⋅ cos⎜ ⎟ ⋅( cos( α) −14⋅ cos ( α) + 17) + 4⋅sin ( α) ⋅cos⎜ ⎟ ⋅( 3− cos( α) ) + 8⋅sin ⎜ ⎟⋅( cos( α) −3) −cos( α) + 14⋅cos( α) −17 ⋅ sin + 1 ⎜ ⎜ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎜ ⎠ ⎝ 2 ⎟ ⎜ ⎠ ⎝ 2 ⎟ ⎠ ⎟ ⎟ ⎝ ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ ⎠ l’area Λ C % è: ( ( ) ( ) ( ( ) ) ( ( ) )) ∞ 2⋅n−1 ⎛ ⎛π −α ⎞ ⎞ 100⋅π ⋅∑⎜ ⋅tan ⎜ ⎟ ⎟ 100 Λ C % = = = = 100⋅ ⋅ ⋅ ⎜ ⎟ ⋅cot ⎜ ⎟ ⋅cot ⎜ ⎟ ⎝ ⎠ 4 ⎝ 2 ⎠ 4 ⎝ 2 ⎠ Nei casi limite: 2 1 2 ⎛ ∞ ⋅n− ⎛ π α ⎞ ⎞ lim C ( α) lim ⎜ ⎛ − ⎞ Λ = π ⋅ tan + + ∑ ⋅ ⎜ ⎟ ⎟=+∞ α→0 α→0 ⎜ ⎜ n= 1 2 ⎝ 4 ⎠ ⎟ ⎝ ⎠ ⎟ ⎝ ⎠ 2 1 2 2 ( ) 1 2 4 ∞ ⋅n− ⋅Λ 100⋅π ⋅ ⋅ f α ⎜ ⎟ C n= ⎝ ⎝ ⎠ ⎠ ⎛α ⎞ ⎛ ⎛π −α ⎞ ⎞ π tan 2 2 T α α ⎜ ⎟ ∑ tan ⎜ ⎟ Λ ⎛ ⎞ ⎛ ⎞ ⎝ 2 ⎠ ⎜ n= 1 ⎝ 4 ⎠ ⎟ 2 1 2 ⎛ ∞ ⋅n− ⎛ π α ⎞ ⎞ lim C ( α) lim C( α) lim ⎜ ⎛ − ⎞ Λ = Λ = π ⋅ tan 0 − + ∑ ⋅ ⎜ ⎟ ⎟= → → α→π⎜ ⎜ n= 1 2 ⎝ 4 ⎠ ⎟ ⎝ ⎠ ⎟ ⎝ ⎠ α π α π 2 40
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