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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❘❯❘❆▲ ❉❖ ❙❊▼■✲➪❘■❉❖ ✲ ❯❋❊❘❙❆
❉✐s❝✐♣❧✐♥❛✿ ❈á❧❝✉❧♦ ◆✉♠ér✐❝♦
Pr♦❢❡ss♦r❡s✿ Pr♦❢✳ ❉r✳ ❖❞❛❝✐r ❆❧♠❡✐❞❛ ◆❡✈❡s ❡ Pr♦❢a ✳ ▼❛✳ ❆♥❣é❧✐❝❛ ❞❡ ❋r❡✐t❛s ❆❧✈❡s
❈✉rs♦✿ ❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛



◆♦çã♦ ❞❡ ❊rr♦s

✶✳✶

■♥tr♦❞✉çã♦

❆ ♦❜t❡♥çã♦ ❞❡ ♠❛ s♦❧✉çã♦ ♥✉♠ér✐❝❛ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❢ís✐❝♦ ♣♦r ♠❡✐♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s
♥✉♠ér✐❝♦s ♥❡♠ s❡♠♣r❡ ❢♦r♥❡❝❡ ✈❛❧♦r❡s q✉❡ s❡ ❡♥❝❛✐①❛♠ ❞❡♥tr♦ ❞❡❧✐♠✐t❡s r❛③♦á✈❡✐s✳ ❊st❛ ❞✐❢❡r❡♥ç❛
❡♥tr❡ ✉♠ ✈❛❧♦r ♥✉♠ér✐❝♦ ❡ ♦ ✈❛❧♦r ❞❛ r❡❛❧ ❞❡ ♠❡❞✐çã♦ é ❝❤❛♠❛❞❛ ❡rr♦ ❡ é ✐♥❡r❡♥t❡ ❛♦ ♣r♦❝❡ss♦✱ ♠✉✐t❛s
✈❡③❡s ♥ã♦ ♣♦❞❡♥❞♦ s❡r ❡✈✐t❛❞♦✳

✲ Pr♦❝❡ss♦ ❞❡ r❡s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❢ís✐❝♦✱ ♣♦r ♠❡✐♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s
♥✉♠ér✐❝♦s
❆❜❛✐①♦ ✉♠ ❡sq✉❡♠❛ ❞❡ r❡s♦❧✉çã♦ q✉❡ ♠♦str❛ ♦s ♣❛ss♦s ♣❛r❛ ❛ r❡s♦❧✉çç❛♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❢ís✐❝♦✳

❉❡ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛✱ t❡♠♦s✿

• ▼♦❞❡❧❛❣❡♠✿

✕ ❋❛s❡ ❞❡♦❜t❡♥çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛
❢ís✐❝♦ ❡♠ q✉❡stã♦✳
• ❘❡s♦❧✉çã♦✿

✕ ❋❛s❡ ❞❡ ♦❜t❡♥çã♦ ❞❛ s♦❧✉çã♦ ❞♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s
♥✉♠ér✐❝♦s✳


✶✳✷

❊rr♦s ♥❛ ❢❛s❡ ❞❡ ♠♦❞❡❧❛❣❡♠

❊①❡♠♣❧♦ ✶✿
P❛r❛ ♦ ❡st✉❞♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ❝♦r♣♦ s✉❥❡✐t♦ ❛ ✉♠❛ ❛❝❡❧❡r❛çã♦ ❝♦♥st❛♥t❡ t❡♠✲s❡ ❛ s❡❣✉✐♥t❡
❡q✉❛çã♦✿
1
d = d0 + v0 t + at2
2

♦♥❞❡✱

✭✶✮

d ✲ ❞✐stâ♥❝✐❛♣❡r❝♦rr✐❞❛❀
d0 ✲ ❞✐stâ♥❝✐❛ ✐♥✐❝✐❛❧❀
v0 ✲ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧❀
t ✲ t❡♠♣♦❀
a ✲ ❛❝❡❧❡r❛çã♦✳

❙✉♣♦♥❞♦✲s❡ q✉❡ ✉♠ ❡♥❣❡♥❤❡✐r♦ q✉❡✐r❛ ❞❡t❡r♠✐♥❛r ❛ ❛❧t✉r❛ ❞❡ ✉♠ ❡❞í✜❝✐♦ ❡ q✉❡ ♣❛r❛ ✐ss♦ ❞✐s♣♦♥❤❛
❛♣❡♥❛s ❞❡ ✉♠❛ ❜♦❧✐♥❤❛ ❞❡ ♠❡t❛✱ ✉♠ ❝r♦♥ô♠❡tr♦ ❡ ❛ ❢ór♠✉❧❛ ❛❝✐♠❛✱ ❡❧❡ s♦❜❡ ❛♦ t♦♣♦ ❞♦ ❡❞✐❢í❝✐♦ ❡
♠❡❞❡ ♦ t❡♠♣♦ q✉❡ ❛ ❜♦❧✐♥❤❛ ❣❛st❛ ♣❛r❛ t♦❝❛r ♦ s♦❧♦✱ ♥❡st❡ ❝❛s♦✱ ✸ s❡❣✉♥❞♦s✳ ▲❡✈❛♥❞♦ ❡st❡ ✈❛❧♦r à
❡q✉❛çã♦ ✶✱ ♦❜té♠✲s❡
d=0+0·3+

1
· 9, 8 · 32 ⇒ d= 44, 1m
2

P❡r❣✉♥t❛✲s❡✿ ❊st❡ r❡s✉❧t❛❞♦ é ❝♦♥✜á✈❡❧❄
❙✉♣♦♥❞♦ q✉❡ ♦ t❡♠♣♦ r❡❛❧ s❡❥❛ ❞❡ ✸✱✺ s❡❣✉♥❞♦s✱ ♦✉ s❡❥❛✱ ✵✱✺ s❡❣✉♥❞♦ ❛ ♠❛✐s✱ ❛ ❛❧t✉r❛ ❞♦ ❡❞✐❢í❝♦
♣❛ss❛ ♣❛r❛ ✻✵♠✦
❈♦♠ ❡st❡ ❡①❡♠♣❧♦ ♣♦❞❡✲s❡ ♥♦t❛r ❛ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ ❞♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❡ ❛ ♣r❡❝✐sã♦ q✉❡ ♦s
❞❛❞♦s ♦❜t✐❞♦s ❡①❡r❝❡♠ s♦❜r❡ ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❛ s♦❧✉ç❛♦ ❡♥❝♦♥tr❛❞❛✳
✶✳✸

❊rr♦s ♥❛ ❢❛s❡ ❞❡ r❡s♦❧✉çã♦

❆ ❛♥á❧✐s❡ ❞♦ ❡rr♦ ♥✉♠ér✐❝♦ ♥♦ ❝♦♠♣✉t❛❞♦r❡stá ❜❛s❡❛❞❛ s♦❜r❡ ❞♦✐s ♣♦♥t♦s✿
✶✮ ❛ ❜❛s❡ ♥✉♠ér✐❝❛ ✉t✐❧✐③❛❞❛ ♥♦ ❝♦♠♣✉t❛❞♦r❀
✷✮ ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❝♦♠♣✉t❛❞♦r ♣♦❞❡ r❡♣r❡s❡♥t❛r ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s✱ ❝❛❞❛
q✉❛❧ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❞í❣✐t♦s✳
❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❞❡♣❡♥❞❡♠✿


• ❉❛ ♣r❡❝✐sã♦ ❞♦s ❞❛❞♦s ❞❡ ❡♥tr❛❞❛❀
• ❉❛ ❢♦r♠❛ ❝♦♠♦ ❡st❡s ❞❛❞♦s sã♦ r❡♣r❡s❡♥t❛❞♦s ♥♦ ❝♦♠♣✉t❛❞♦r❀
• ❉❛s ♦♣❡r❛çõ❡s ❡❢❡t✉❛❞❛s✳

❖s ❞❛❞♦s ❞❡ ❡♥tr❛❞❛♣♦ss✉❡♠ ✉♠❛ ✐♠♣r❡❝✐sã♦✱ ❡♠ ♠✉✐t♦s ❝❛s♦s✱ ✐♥❡✈✐tá✈❡❧ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦
♠❡❞✐❞❛s ❞❡ ❝♦rr❡♥t❡ ❡❧étr✐❝❛✳
❯♠ ❡①❡♠♣❧♦ ❞❡ ❡rr♦ ♥❛ r❡s♦❧✉çã♦ é ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❝✐r❝✉❧❛r ♦♥❞❡ ♦ r❛✐♦ é ✐❣✉❛❧
❛ ✶✵✵ m✳
❙❛❜❡♥❞♦ q✉❡ ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❝✐r❝✉❧❛r é ❞❛❞❛ ♣♦r A = π ·r2 ✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r três s✐t✉❛çõ❡s
♣❛rt✐❝✉❧❛r❡s✿
❛✮ ❙❡ π = 3, 14✱ ❡♥tã♦ A1 = 31.400 m2
❜✮ ❙❡ π = 3, 1416✱ ❡♥tã♦ A2 = 31.416 m2
❝✮ ❙❡ π =3, 141592654✱ ❡♥tã♦ A3 = 31.415, 92654 m2
❙❡rá q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r ❛ ár❡❛ ❡①❛t❛❄
P❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s✱ ♠✉✐t❛s ✈❡③❡s é ♥❡❝❡ssár✐♦ r❡❝♦rr❡r ❛ ❝❡rt❛s ❛♣r♦①✐♠❛çõ❡s
q✉❡ ♣♦❞❡♠ ✈✐r ❛ ❣❡r❛r ❡rr♦s ❞✉r❛♥t❡ ❛ ❝♦♥✈❡rsã♦ ❞❡ ❜❛s❡s ❝♦♠♦ ♠♦str❛r❡♠♦s ❛ s❡❣✉✐r✳

✶✳✸✳✶

❘❡♣r❡s❡♥t❛ç❛♦ ♥✉♠ér✐❝❛ ✲ ❈♦♥✈❡rsã♦ ❞❡ ❜❛s❡s

◗✉❛♥❞♦ ❡s❝r❡✈❡♠♦s ♥❛ r❡♣r❡s❡♥t❛çã♦ ♥✉♠ér✐❝❛ ✉s✉❛❧ ✷✶✸✱✶✹✶✺✱ ❡st❛♠♦s ✉t✐❧✐③❛♥❞♦✐♠♣❧✐❝✐t❛♠❡♥t❡
✉♠ s✐st❡♠❛ ❞❡ ♣♦s✐❝✐♦♥❛♠❡♥t♦ ♥❛ ❜❛s❡ ✶✵✱ ♦✉ s❡❥❛✱
213, 1415 = 2 × 102 + 1 × 101 + 3 × 100 + 1 × 10−1 + 4 × 10−2 + 1 × 10−3 + 5 × 10−4

❉❡ ♠♦❞♦ ❣❡r❛❧✱ ✉♠ ♥ú♠❡r♦ α✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r αm , αm−1 , αm−2 , · · · , α2 , α1 , α0 , α−1 , α−2 , · · · , αn ✱
♥♦ q✉❛❧ αi ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} é ❡q✉✐✈❛❧❡♥t❡ ❛ s♦♠❛
m

αi · 10i

α=
i=−n

P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♦ ♥ú♠❡r♦ ✷✶✸✱✶✹✶✺✱ t❡♠♦s✿
m = 2✱ n = −4
α2 = 2✱ α1= 1✱ α2 = 3✱ α−1 = 1✱ α−2 = 4✱ α−3 = 1✱ α−4 = 5✳

✶✳✸✳✷

❈♦♥✈❡rsã♦ ❞❡ ❜✐♥ár✐♦ ♣❛r❛ ♥ú♠❡r♦ ❞❡❝✐♠❛❧

❯♠ ♥ú♠❡r♦ ♥❛ ❜❛s❡ ✷ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿
am 2m + am−1 2m−1 + · · · + a2 22 + a1 21 + a0 20 + a−1 2−1 + · · · + an 2n ♦✉ ❛✐♥❞❛



m

= ai 2i
i=n

♦♥❞❡ ai = 0 ♦✉ ✶ ❡ n, m sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ n ≤ 0 ❡ m ≥ 0

❊①❡♠♣❧♦s✿
10112 = 1 × 23 + 0 × 22 + 1 ×...
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