Arquimedes: um ponto de apoio para o método científico

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❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

❆rq✉✐♠❡❞❡s✿ ❯♠ P♦♥t♦ ❞❡ ❆♣♦✐♦ ♣❛r❛ ♦

▼ét♦❞♦ ❈✐❡♥tí❢✐❝♦

❍❡r♠❡s ❆♥tô♥✐♦ P❡❞r♦s♦

Pr♦❢❡ss♦r ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ❯◆❊❙P✴■❇■▲❈❊ ✕ ❈❛♠♣✉s ❞❡ ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦

❤❡r♠❡s❅✐❜✐❧❝❡✳✉♥❡s♣✳❜r

❘❡s✉♠♦

❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭✷✽✼ ✕ ✷✶✷ ❛✳❈✳✮✱ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❛✐♦r ♠❛t❡♠át✐❝♦ ❞❛ ❆♥✲ t✐❣✉✐❞❛❞❡✱ ❛♣❡r❢❡✐ç♦♦✉ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❛tr✐❜✉í❞♦ ❛ ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽ ✕ ✸✺✺ ❛✳❈✳✮✳ ❊st❡ ♠ét♦❞♦ s❡ t♦r♥♦✉ ♦ ♠♦❞❡❧♦ ❣r❡❣♦ ❡ ❞♦ ❘❡♥❛s❝✐♠❡♥t♦ ♥❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s✳ ❊r❛ ♠✉✐t♦ r✐❣♦r♦s♦✱ ♠❛s t✐♥❤❛ ❛ ❞❡s✈❛♥t❛❣❡♠ ❞❡ ♦ r❡s✉❧t❛❞♦✱ ♣❛r❛ s❡r ♣r♦✈❛❞♦✱ ♣r❡❝✐s❛r s❡r ❝♦♥❤❡❝✐❞♦ ❛♥t❡s✳

❊①✐st❡♠ ✐♥❞✐❝❛çõ❡s ❝❧❛r❛s ❞❡ q✉❡ ✉♠ ♦✉tr♦ ♠ét♦❞♦ t❛♠❜é♠ ❡r❛ ✉t✐❧✐③❛❞♦✳ ◆✉♠❛ ❝❛rt❛ ❛ ❊r❛tóst❡♥❡s ✭✷✼✻ ✕ ✶✾✻ ❛✳❈✳✮✱ q✉❡ ♥ã♦ t✐♥❤❛ s✐❞♦ ❞❡s❝♦❜❡rt❛ ❛té ✶✾✵✻✱ ❆rq✉✐♠❡❞❡s ❢❛③ r❡✈❡❧❛çõ❡s ❞❡ ❝♦♠♦ ❝❤❡❣❛r❛ ❛♦s r❡s✉❧t❛❞♦s ✉t✐❧✐③❛♥❞♦ ❛❧❛✈❛♥❝❛s ♣❛r❛ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ✜❣✉r❛s ❣❡♦♠étr✐❝❛s✳ ❖ r❡s✉♠♦ ❞♦ ❛rt✐❣♦ ❡♠ ♣♦rt✉❣✉ês ❞❡✈❡ ✈✐r ❛q✉✐✳ P❡❞❡✲s❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢r❛s❡s ❝♦♥s❝✐s❛s ❡ ♦❜❥❡t✐✈❛s ✭♥ã♦ ✉♠❛ s✐♠♣❧❡s ❡♥✉♠❡r❛çã♦ ❞❡ tó♣✐❝♦s✮ q✉❡ ♥ã♦ ✉❧tr❛♣❛ss❡ ✸✵✵ ♣❛❧❛✈r❛s✳

P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ❆rq✉✐♠❡❞❡s✱ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦✱ ➪r❡❛s ❡ ❱♦❧✉♠❡s✳

❆r❝❤✐♠❡❞❡s✿ ❛ ❢✉❧❝r✉♠ ❢♦r t❤❡ s❝✐❡♥t✐❢✐❝ ♠❡t❤♦❞

❆❜str❛❝t

❆r❝❤✐♠❡❞❡s ♦❢ ❙②r❛❝✉s❡ ✭✷✽✼ ✕ ✷✶✷ ❇❈✮ ✐s r❡❣❛r❞❡❞ ❛s t❤❡ ❣r❡❛t❡st ♠❛t❤❡♠❛t✐❝✐❛♥ ✐♥ ❝❧❛ss✐❝❛❧ ❛♥t✐q✉✐t②✳ ❍❡ ✐♠♣r♦✈❡❞ t❤❡ ♠❡t❤♦❞ ♦❢ ❡①❤❛✉st✐♦♥ ❛ttr✐❜✉t❡❞ t♦ ❊✉❞♦①✉s ♦❢ ❈♥✐❞✉s ✭✹✵✽ ✕ ✸✺✺ ❇❈✮✳ ❚❤✐s ♠❡t❤♦❞ ❤❛s ❜❡❝♦♠❡ t❤❡ ●r❡❡❦ ❛♥❞ ❘❡♥❛✐ss❛♥❝❡ ♠♦❞❡❧ ✐♥ ❞❡♠♦♥str❛t✐♦♥s ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❛r❡❛s ❛♥❞ ✈♦❧✉♠❡s✳ ❚❤♦✉❣❤ ✐t ✇❛s ✈❡r② r✐❣♦r♦✉s✱ t❤❡ ❞✐s❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s ♠❡t❤♦❞ ✇❛s t❤❡ ❢❛❝t t❤❛t t❤❡ r❡s✉❧t ❤❛❞ t♦ ❜❡ ❦♥♦✇♥ ❜❡❢♦r❡ ❜❡✐♥❣ ♣r♦✈❡❞✳

❚❤❡r❡ ❛r❡ s♦♠❡ ❝❧❡❛r ✐♥❞✐❝❛t✐♦♥s t❤❛t ❛♥♦t❤❡r ♠❡t❤♦❞ ✇❛s ❛❧s♦ ✉s❡❞✳ ■♥ ❛ ❧❡tt❡r t♦ ❊r❛t♦st❤❡♥❡s ✭✷✼✻ ✕ ✶✾✻ ❇❈✮✱ ✇❤✐❝❤ ❤❛❞ ♥♦t ❜❡❡♥ ❞✐s❝♦✈❡r❡❞ ✉♥t✐❧ ✶✾✵✻✱ ❆r❝❤✐♠❡❞❡s r❡✈❡❛❧s ❤♦✇ ❤❡ ②✐❡❧❞❡❞ t❤❡ r❡s✉❧ts ✉s✐♥❣ ❧❡✈❡rs t♦ ❡st❛❜❧✐s❤ t❤❡ ❜❛❧❛♥❝❡ ♦❢ ❣❡♦♠❡tr✐❝ ❢♦r♠s✳

❑❡②✇♦r❞s✿ ❆r❝❤✐♠❡❞❡s✱ ▼❡t❤♦❞ ♦❢ ❊①❤❛✉st✐♦♥✱ ❆r❡❛s ❛♥❞ ❱♦❧✉♠❡s✳

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✶ ❆rq✉✐♠❡❞❡s ❡ s❡✉ t❡♠♣♦

❆rq✉✐♠❡❞❡s ♥❛s❝❡✉ ❡ ✈✐✈❡✉ ❡♠ ❙✐r❛❝✉s❛✱ ✉♠❛ ❝✐❞❛❞❡ ❞❛ ❙✐❝í❧✐❛ q✉❡ ❡①✐st❡ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳ ❈♦♥st❛ q✉❡ ❡❧❡ ♠♦rr❡✉ ♥♦ ❛♥♦ ✷✶✷ ❛✳❈✳ ❝♦♠ ❛ ✐❞❛❞❡ ❞❡ ✼✺ ❛♥♦s ❡ ❞❛í s❡ ❝♦♥❝❧✉✐ q✉❡ ❡❧❡ ♥❛s❝❡✉ ♥♦ ❛♥♦ ❞❡ ✷✽✼ ❛✳❈✳

❙✐r❛❝✉s❛ ❡r❛ ❝✐❞❛❞❡✲❡st❛❞♦ ❞❛s ♠✉✐t❛s q✉❡ ♦s ❣r❡❣♦s ❢✉♥❞❛r❛♠✱ ♣♦rt❛♥t♦ ❆rq✉✐♠❡❞❡s ❡r❛ ✉♠ ♠❛t❡♠át✐❝♦ ❣r❡❣♦✳ ▼❛s ♥❡ss❛ é♣♦❝❛ ❛ ●ré❝✐❛ ❥á ❤❛✈✐❛ s✐❞♦ ❝♦♥q✉✐st❛❞❛ ♣♦r ❆❧❡①❛♥❞r❡ ❞❛ ▼❛❝❡❞ô♥✐❛✱ q✉❡ ❡①♣❛♥❞✐r❛ s❡✉ ■♠♣ér✐♦ ♣❡❧❛ ➪s✐❛ ❡ ❊❣✐t♦✳ ❆❧❡①❛♥❞r❡ r❡s♦❧✈❡r❛ ✐♥st❛❧❛r ❛ ❝❛♣✐t❛❧ ❞♦ ■♠♣ér✐♦ ♥✉♠❛ ❝✐❞❛❞❡ ❛ s❡r ❝♦♥str✉í❞❛ ♥♦ ❡①tr❡♠♦ ♦❡st❡ ❞♦ ❞❡❧t❛ ❞♦ r✐♦ ◆✐❧♦✳ ■st♦ ❢♦✐ ❢❡✐t♦✱ ♥ã♦ ♣♦r ❆❧❡①❛♥❞r❡✱ q✉❡ ♠♦rr❡✉ ❡♠ ✸✷✸ ❛✳❈✳✱ ♠❛s ♣♦r ✉♠ ❞♦s s❡✉s ❣❡♥❡r❛✐s✱ Pt♦❧♦♠❡✉ ❙♦t❡r✱ q✉❡ ✜❝♦✉ ❝♦♠ ❛ ♣❛rt❡ ❡❣í♣❝✐❛ ❞♦ ■♠♣ér✐♦ ❡ ✐♥✐❝✐♦✉ ✉♠❛ ❞✐♥❛st✐❛ ❣r❡❣❛ ♥♦ ❊❣✐t♦✳ ❆ss✐♠ s✉r❣✐✉ ❆❧❡①❛♥❞r✐❛✱ q✉❡ s❡ t♦r♥♦✉ ✉♠ ❝❡♥tr♦ ❢❛♠♦s♦ ❞❛ ❝✉❧t✉r❛ ❝❤❛♠❛❞❛ ✏❤❡❧❡♥íst✐❝❛✑ ❡ q✉❡ ❝♦♥t❛✈❛ ❛té ❝♦♠ ✉♠❛ ✈❡r❞❛❞❡✐r❛ ✉♥✐✈❡rs✐❞❛❞❡ ✕ ✉♠ ✐♥st✐t✉t♦ ❞❡ ❛❧t♦s ❡st✉❞♦s ❡ ✉♠❛ ❜✐❜❧✐♦t❡❝❛ ♠✉✐t♦ ❢❛♠♦s❛✱ q✉❡ ❝❤❡❣♦✉ ❛ t❡r ✼✺✵✵✵✵ ✈♦❧✉♠❡s✳ ❊♠ ❆❧❡①❛♥❞r✐❛✱ ❛ ▼❛t❡♠át✐❝❛ ♦❝✉♣❛✈❛ ✉♠ ❧✉❣❛r ❞❡ ❞❡st❛q✉❡ ❡ ♥♦♠❡s ❝♦♠♦ ❊✉❝❧✐❞❡s✱ ❆♣♦❧ô♥✐♦✱ ❆rq✉✐♠❡❞❡s✱ ❊r❛tóst❡♥❡s✱ ❆r✐st❛r❝♦ ❡ Pt♦❧♦♠❡✉ ✭♦ ❛strô♥♦♠♦✱ s❡♠ ♥❡♥❤✉♠ ♣❛r❡♥t❡s❝♦ ❝♦♠ ♦s r❡✐s Pt♦❧♦♠❡✉s✮ ♣❡rt❡♥❝❡r❛♠ à ❊s❝♦❧❛ ❞❡ ❆❧❡①❛♥❞r✐❛✳ ➱ ✈❡r❞❛❞❡ q✉❡ ❆rq✉✐♠❡❞❡s ✈✐✈❡✉ ❡♠ ❙✐r❛❝✉s❛✱ ♠❛s ❡st✉❞♦✉ ❡♠ ❆❧❡①❛♥❞r✐❛ ❡ ♠❛♥t✐♥❤❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠ ✈ár✐♦s sá❜✐♦s ❞❡ ❧á✱ ❝♦♠♦ ❊r❛tóst❡♥❡s✳ ❊st❡ ú❧t✐♠♦ ❡r❛ ❜✐❜❧✐♦t❡❝ár✐♦✱ ✉♠ ❤♦♠❡♠ ❞❡ s❛❜❡r ✉♥✐✈❡rs❛❧✱ ❜❡♠ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ ❝❤❛♠❛❞♦ ✏❝r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s✑✱ ♠❛s s❡✉ ❢❡✐t♦ ♠❛✐s ♥♦tá✈❡❧ ❢♦✐ ❝❛❧❝✉❧❛r ♦ r❛✐♦ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❛ ❚❡rr❛✳

◆❛ é♣♦❝❛ ❡♠ q✉❡ ✈✐✈❡✉ ❆rq✉✐♠❡❞❡s✱ ❘♦♠❛ ❥á ❡st❛✈❛ ❡♠ ❡①♣❛♥sã♦✱ ❝♦♠ ♠✉✐t❛s ❣✉❡rr❛s ❞❡ ❝♦♥q✉✐st❛s✱ ❞❡♥tr❡ ❛s q✉❛✐s sã♦ ❜❡♠ ❝♦♥❤❡❝✐❞❛s ❛s ❝❤❛♠❛❞❛s ✏❣✉❡rr❛s ♣ú♥✐❝❛s✑ ❝♦♥tr❛ ❈❛rt❛❣♦✳ ❊st❛ ❝✐❞❛❞❡ ✜❝❛✈❛ ♦♥❞❡ é ❤♦❥❡ ✉♠ s✉❜úr❜✐♦ ❞❡ ❚✉♥✐s✱ ❛ ❝❛♣✐t❛❧ ❞❛ ❚✉♥ís✐❛✳ ◆❛q✉❡❧❡ t❡♠♣♦✱ ❈❛rt❛❣♦ ❝♦♥tr♦❧❛✈❛ ✉♠❛ ❡①t❡♥s❛ r❡❣✐ã♦ q✉❡ s❡ ❡st❡♥❞✐❛ ❛té ❛ ❊s♣❛♥❤❛✱ ❝♦♥st✐t✉✐♥❞♦✲ s❡ ♥✉♠❛ ✐♥❝ô♠♦❞❛ r✐✈❛❧ ❞❡ ❘♦♠❛✳ ◆❛ s❡❣✉♥❞❛ ❞❛s ❣✉❡rr❛s ♣ú♥✐❝❛s✱ ❙✐r❛❝✉s❛ s❡ ❛❧✐❛r❛ ❛ ❈❛rt❛❣♦✱ ❞❛í t❡r s♦❢r✐❞♦ ✉♠❛ ✐♥✈❡st✐❞❛ ❢❛t❛❧ ❞❡ ❘♦♠❛✳ ❍á ✐♥❞í❝✐♦s ❞❡ q✉❡ ❙✐r❛❝✉s❛ r❡s✐st✐✉ ❜r❛✈❛♠❡♥t❡ ❛♦s ❛t❛q✉❡s ❞♦ ❣❡♥❡r❛❧ ▼❛r❝❡❧♦✱ ❣r❛ç❛s às ♠áq✉✐♥❛s ❞❡ ❣✉❡rr❛ ✐❞❡❛❧✐③❛❞❛s ♣♦r ❆rq✉✐♠❡❞❡s❀ ♠❛s ❞❡♣♦✐s ❞❡ ✉♠ ❧♦♥❣♦ ❝❡r❝♦ ❛❝❛❜♦✉ ♣♦r s✉❝✉♠❜✐r à s✉♣❡r✐♦r✐❞❛❞❡ ❞❛s tr♦♣❛s r♦♠❛♥❛s✳ ❍á ✈ár✐❛s ✈❡rsõ❡s s♦❜r❡ ❛ ♠♦rt❡ ❞❡ ❆rq✉✐♠❡❞❡s❀ s❡❣✉♥❞♦ ✉♠❛ ❞❡❧❛s✱ ❞✉r❛♥t❡ ♦ s❛q✉❡ ❞❛ ❝✐❞❛❞❡✱ ❡♠ ✷✶✷ ❛✳❈✳✱ ❡❧❡ ❢♦✐ ♠♦rt♦ ♣♦r ✉♠ s♦❧❞❛❞♦ r♦♠❛♥♦✱ q✉❛♥❞♦ ❛❜s♦rt♦✱ s❡ ♦❝✉♣❛✈❛ ❝♦♠ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s✳

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❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

❋✐❣✉r❛ ✶✿ ❆ ♠♦rt❡ ❞❡ ❆rq✉✐♠❡❞❡s ❛ ♣❛rt✐r ❞❡ ✉♠❛ ♣✐♥t✉r❛ ❞❡ ●✳❈✳❊✳ ❈♦✉rt♦✐s

❋✐❣✉r❛ ✷✿ ❆ ♠♦rt❡ ❞❡ ❆rq✉✐♠❡❞❡s✱ ❋♦t♦ ❚❤❡ ▼❛♥s❡❧ ❈♦❧❧❡❝t✐♦♥ ✲ ▲♦♥❞r❡s

❆rq✉✐♠❡❞❡s ❡r❛ ❜❡♠ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ r❡✐ ❍✐❡rã♦ ❞❡ ❙✐r❛❝✉s❛ ❡ t❛❧✈❡③ ❢♦ss❡ s❡✉ ♣❛r❡♥t❡✳ ❈♦♥t❛✲s❡ q✉❡ ❍✐❡rã♦ ♠❛♥❞♦✉ ❢❛③❡r ✉♠❛ ❝♦r♦❛ ❞❡ ♦✉r♦✱ ♠❛s t❡✈❡ r❛③õ❡s ♣❛r❛ ❞❡s❝♦♥✜❛r ❞❡ q✉❡ ♦ ♦✉r♦ ❞❛ ❝♦r♦❛ ❤♦✉✈❡ss❡ s✐❞♦ ♠✐st✉r❛❞♦ ❝♦♠ ♠✉✐t❛ ♣r❛t❛✳ ❊❧❡ ❝♦♠✉♥✐❝♦✉ ♦ ❢❛t♦ ❛ ❆rq✉✐♠❡❞❡s✱ ♣❛r❛ q✉❡ ♦ sá❜✐♦ ❡♥❝♦♥tr❛ss❡ ✉♠ ♠❡✐♦ ❞❡ ❞✐r✐♠✐r s✉❛s ❞ú✈✐❞❛s✳ ❉✐③ ❛ ❤✐stór✐❛ q✉❡ ❆rq✉✐♠❡❞❡s ❞❡s❝♦❜r✐✉ ❝♦♠♦ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❡♥q✉❛♥t♦ t♦♠❛✈❛ ❜❛♥❤♦ ❡ r❡✢❡t✐❛ s♦❜r❡ ♦ ❢❛t♦ ❞❡ q✉❡ ♦s ❝♦r♣♦s ✐♠❡rs♦s ♥❛ á❣✉❛ ✕ ❝♦♠♦ s❡✉ ♣ró♣r✐♦ ❝♦r♣♦ ✕ s❡ t♦r♥❛♠ ♠❛✐s ❧❡✈❡s✱ ❡①❛t❛♠❡♥t❡ ♣❡❧♦ ♣❡s♦ ❞❛ á❣✉❛ q✉❡ ❞❡s❧♦❝❛♠✳ ❊st❡ ❢❛t♦ ❧❤❡ t❡r✐❛ ♣❡r♠✐t✐❞♦ ✐❞❡❛❧✐③❛r ✉♠ ♠♦❞♦ ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦r♦❛✱ ❡ tã♦ ❡①❝✐t❛❞♦ ❡❧❡ t❡r✐❛ ✜❝❛❞♦ ❝♦♠ ❛ ❞❡s❝♦❜❡rt❛ q✉❡ s❛✐✉ ♥✉ ♣❡❧❛s r✉❛s ❞❡ ❙✐r❛❝✉s❛ ❣r✐t❛♥❞♦ ✏❊✉r❡❦❛✦ ❊✉r❡❦❛✑✱ q✉❡ s✐❣♥✐✜❝❛ ✏❉❡s❝♦❜r✐✦ ❉❡s❝♦❜r✐✦✑✳

❋✐❣✉r❛ ✸✿ ❆rq✉✐♠❡❞❡s ♥♦ ❜❛♥❤♦✱ ❣r❛✈✉r❛ ❞❛ ♦❜r❛ ❞❡ ●❛✉❧t❤❡r✉s ❘✐✈✐✉s✱ ◆✉r❡♠❜❡r❣✱ ✶✺✼✹

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✷ ❖s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s ✭❡♠ ♦r❞❡♠ ❝r♦♥♦❧ó❣✐❝❛

♣r♦✈á✈❡❧✮

• ❙♦❜r❡ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✱ ■✳

• ❆ q✉❛❞r❛t✉r❛ ❞❛ ♣❛rá❜♦❧❛✳

• ❙♦❜r❡ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✱ ■■✳

• ❙♦❜r❡ ❛ ❡s❢❡r❛ ❡ ♦ ❝✐❧✐♥❞r♦✱ ■✱ ■■✳

• ❙♦❜r❡ ❛s ❡s♣✐r❛✐s✳

• ❙♦❜r❡ ♦s ❝♦♥❡s ❡ ❡s❢❡ró✐❞❡s✳

• ❙♦❜r❡ ♦s ❝♦r♣♦s ✢✉t✉❛♥t❡s ■✱ ■■✳

• ❆ ♠❡❞✐❞❛ ❞♦ ❝ír❝✉❧♦✳

• ❖ ❈♦♥t❛❞♦r ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛✳

• ❆ ❝❛rt❛ ❛ ❊r❛tóst❡♥❡s s♦❜r❡ ♦ ▼ét♦❞♦✳

❆ s❡❣✉✐r ❢❛r❡♠♦s ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s q✉❡ ❝♦♥s✐❞❡r❛♠♦s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❛❧❣✉♠❛s ♦❜r❛s ❞❡ ❆rq✉✐♠❡❞❡s✳

✸ ❖ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦

❖ ♣r✐♠❡✐r♦ ♠ét♦❞♦ ✉s❛❞♦ ♥♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✱ ❤♦❥❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦✱ ❢♦✐ ❝r✐❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦✭✹✵✽✲✸✺✺ ❛✳❈✳✮✱ ✉s❛❞♦ ❡ ❛♣❡r❢❡✐ç♦❛❞♦ ♣♦r ❊✉❝❧✐❞❡s✭❝✳✸✵✵ ❛✳❈✳✮ ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛✭✷✽✼✲✷✶✷ ❛✳❈✳✮✳ ❖ ♠ét♦❞♦ ❜❛s❡✐❛✲s❡ ♥❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

❙❡ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ q✉❛❧q✉❡r s✉❜tr❛✐✲s❡ ✉♠❛ ♣❛rt❡ ♥ã♦ ♠❡♥♦r q✉❡ s✉❛ ♠❡t❛❞❡✱ ❞♦ r❡st❛♥t❡ s✉❜tr❛✐✲s❡ t❛♠❜é♠ ✉♠❛ ♣❛rt❡ ♥ã♦ ♠❡♥♦r q✉❡ s✉❛ ♠❡t❛❞❡✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❝❤❡❣❛r✲s❡✲á ❡♠ ❛❧❣✉♠❛ ❡t❛♣❛ ❞❡ss❡ ♣r♦❝❡ss♦ ❛ ✉♠❛ ❣r❛♥❞❡③❛ ♠❡♥♦r q✉❡ q✉❛❧q✉❡r ❣r❛♥❞❡③❛ ❞❛ ♠❡s♠❛ ❡s♣é❝✐❡ ✜①❛❞❛ ♣r❡✈✐❛♠❡♥t❡✳

✭❆ ♣r♦✈❛ ❡♥❝♦♥tr❛✲s❡ ♥❛ Pr♦♣♦s✐çã♦ ❳✲✶✱ ❞❡ ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✮

(5)

❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

Pr♦❝✉r❛♠♦s ✐❧✉str❛r ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❝♦♠ ❛s ✜❣✉r❛s ❛❝✐♠❛✱ r❡❢❡r❡♥t❡s ❛ ❞♦✐s tr❛❜❛✲ ❧❤♦s ✐♠♣♦rt❛♥t❡s q✉❡ ♠❡r❡❝❡r❛♠ ❛t❡♥çã♦ ❡s♣❡❝✐❛❧ ❞❡ ❆rq✉✐♠❡❞❡s✿ ❆ ♠❡❞✐❞❛ ❞♦ ❈ír❝✉❧♦ ❡ ❆ ◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛✳

✹ ❆ ♠❡❞✐❞❛ ❞♦ ❝ír❝✉❧♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❆rq✉✐♠❡❞❡s ♣r♦✈❛ três ♣r♦♣♦s✐çõ❡s✿

✶✳ ❚♦❞♦ ❝ír❝✉❧♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡♠ q✉❡ ♦s ❝❛t❡t♦s sã♦ ✐❣✉❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦ r❛✐♦ ❡ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦ ❝ír❝✉❧♦✳

Pr♦✈❛✿

❙❡❥❛♠ r ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦✱ c ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ C ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡T

❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳ ❚❡♠♦s q✉❡ ♣r♦✈❛r C =T✳

(6)

✐✳ ❙✉♣♦♥❤❛ C > T✳ ❙❡❥❛ A= C✕T✱ A > 0✳ ❈♦♥s✐❞❡r❡ ✉♠

♣♦❧í❣♦♥♦ r❡❣✉❧❛r ✐♥s❝r✐t♦ ❞❡ ❛♣ót❡♠❛ m✬✱ ♣❡rí♠❡tr♦p✬ ❡

❞❡ ár❡❛ P✬✱ t❛❧ q✉❡ C✕P✬ < A✳ ❆ss✐♠✱ C✕P✬ < A =

C✕T✱ ♦✉ s❡❥❛✱P✬ > T✳ ▼❛s P′ ₌ p ′_m′

2 ❡T

′ ₌ c.r

2 ✱ ❧♦❣♦

p✬m✬ > cr✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s p✬ < c ❡ m✬ < r✳

❊♥tã♦ C ≤T✳

✐✐✳ ❙✉♣♦♥❤❛C < T✳ ❙❡❥❛A=T✕C❡ ❝♦♥s✐❞❡r❡ ✉♠ ♣♦❧í❣♦♥♦

❝✐r❝✉♥s❝r✐t♦ ❞❡ ❛♣ót❡♠❛ r✱ ♣❡rí♠❡tr♦ p ❡ ár❡❛ P ❝♦♠ P✕C < A✳ ❆ss✐♠✱ P✕C < A=T✕C✱ ♦✉ s❡❥❛✱ p < T✱ ♦✉

❛✐♥❞❛✱ rp

2 <

rc

2✱ ✐st♦ é✱ p < c✱ ❛❜s✉r❞♦✳ ❊♥tã♦✱ C ≥ T✳

P♦rt❛♥t♦✱ C =T = cr 2.

✷✳ ❙❡ cé ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ d é ♦ ❞✐â♠❡tr♦ ❡♥tã♦

3 + 10 71

d < c <

3 + 10 70

d, ou seja,

3 + 10

71 < π <3 + 1 7.

❊♠ ❞❡❝✐♠❛✐s t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿

3,14084< π <3,142858

✸✳ ❖ ❝ír❝✉❧♦ ❡stá ♣❛r❛ ♦ q✉❛❞r❛❞♦ ❞❡ s❡✉ ❞✐â♠❡tr♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♥❛ r❛③ã♦ 11 14.

✺ ❆ q✉❛❞r❛t✉r❛ ❞❛ ♣❛rá❜♦❧❛

❱❡❥❛♠♦s ❝♦♠♦ ❆rq✉✐♠❡❞❡s ❞❡♠♦♥str♦✉ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ s❡❣✲ ♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ é 4

3 ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ❞❡ ♠❡s♠❛ ❜❛s❡ ❡ ❛❧t✉r❛✳

(7)

❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ✜❣✉r❛ ❛❝✐♠❛ r❡♣r❡s❡♥t❡ ✉♠❛ ♣♦rçã♦ ❞❡ ♣❛rá❜♦❧❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ❝♦r❞❛C✬C✱ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ s❡✉ ❡✐①♦AB✳ ❈♦♠♦ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝♦♥✲

❥✉♥t♦ ❞♦s ♣♦♥t♦sP t❛✐s q✉❡AP✬ s❡❥❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛ (P✬P)2

✐st♦ é✱ ❡♠ ♥♦t❛çã♦ ♠♦❞❡r♥❛✱y=

kx2

.

❆rq✉✐♠❡❞❡s ♠♦str♦✉ q✉❡ ❡ss❛ ♣♦rçã♦ ❞❡ ♣❛rá❜♦❧❛ é 4

3 ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ C✬AC✱ ♦ q✉❡

❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡ ❛ ár❡❛ ❧✐♠✐t❛❞❛ ♣♦r AB✱ BC✱ ❡ ❛ ♣❛rá❜♦❧❛ é 4

3 ❞❛ ár❡❛ ❞❡ ABC✳ P❛r❛

t❛♥t♦ ❡❧❡ ✏❡①❛✉r✐✉✑ ❛ ár❡❛ ♣❛r❛❜ó❧✐❝❛ s♦♠❛♥❞♦ ♣r✐♠❡✐r♦ ♦ tr✐â♥❣✉❧♦ADC ❛♦ABC✱ ♦♥❞❡ Dé

♦ ♣♦♥t♦ ❡♠ q✉❡ ✉♠❛ ♣❛r❛❧❡❧❛ ❛AB♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦M ❞❡BC ❝♦rt❛ ❛ ♣❛rá❜♦❧❛✱ ❡ ♠♦str❛♥❞♦

q✉❡ADC = 1

4ABC.❆ s❡❣✉✐r ❝♦♥str✉✐✉ ♣❛r❛❧❡❧❛s ❛AB ♣♦rM✬❡M✑✱ ♣♦♥t♦s ♠é❞✐♦s ❞❡M C ❡

BM✱ ❛s q✉❛✐s ❝♦rt❛♠ ❛ ♣❛rá❜♦❧❛ ❡♠D✬ ❡D✑❀ ❡♥tã♦ ♠♦str♦✉ q✉❡AD✑D+DD✬C = 1

4ADC = 1

42ABC✳ ❈♦♥t✐♥✉❛♥❞♦ ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ❝♦♠ ❡st❡ ♣r♦❝❡ss♦✱ ❝❤❡❣❛✲s❡ à ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ❛ ár❡❛

♣❛r❛❜ó❧✐❝❛ é ❞❛❞❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♣♦r

ABC+ 1

4ABC + 1

42ABC+...+

1

4nABC, (1)

❆ q✉❛❧✱ à ♠❡❞✐❞❛ q✉❡ n ❝r❡s❝❡✱ ❛♣r♦①✐♠❛✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡ 4

3ABC✳

❆ ♣r♦✈❛ ❞❡ q✉❡ ADC = 1

4ABC ❢❛③✲s❡ ❝♦♠♦ s❡ s❡❣✉❡✱ ❝♦♠ ❛ ♥♦t❛çã♦ ❡ ♦s s❡❣♠❡♥t♦s

❝♦♥str✉í❞♦s ❞❛ ✜❣✉r❛✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛✱ AF = k(F D)2 ❡ AB = k(BC)2. ❈♦♠♦ F D = BM = 1

2BC, ❞❡❞✉③✲s❡ q✉❡ AF = HD = 1

4AB✳ P♦r s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱

EM AB =

M C BC =

1

2, ❞❡ ♠♦❞♦ q✉❡EM = 1

2AB✳ ❉❛í

DE =AB−HD−EM =AB − 1

4AB − 1 2AB =

1 4AB.

❆ss✐♠✱ ADE ❡ AEM t❡♠ ❛ ♠❡s♠❛ ❛❧t✉r❛ AH ❡ ❜❛s❡s DE = 1

4AB ❡ EM = 1 2AB,

(8)

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ▲♦❣♦✱ ADE = 1

2AEM✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ DEC = 1

2EM C✱ ❞❡ ♠❛♥❡✐r❛ q✉❡✱

♣♦r ❛❞✐çã♦✱ ADC = 1

2ACM✳ ❆❧é♠ ❞✐ss♦✱ ACM ❡ AM B t❡♠ ❜❛s❡s ✐❣✉❛✐s (M C e BM)❡

♠❡s♠❛ ❛❧t✉r❛ (AB), ❡ ❛ss✐♠ADC = 1 4ABC.

❆♥❛❧♦❣❛♠❡♥t❡✱ ❝♦♠ ♦ ✉s♦ ❞♦s s❡❣♠❡♥t♦s ❝♦♥str✉í❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❛ ✜❣✉r❛✱ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡

DD′_C ₌ 1

4DCE e AD”D= 1

4ADE, ❞❡ ❢♦r♠❛ q✉❡

Ad”D+DD′_C ₌ 1

4ADC = 1 42ABC,

❝♦♠♣❧❡t❛♥❞♦ ❛ss✐♠ ❛ s❡❣✉♥❞❛ ❡t❛♣❛ ❞❛ ♣r♦✈❛✳

❈♦♠♦ ❞❡❝♦rrê♥❝✐❛ ❞❛ ◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛✱ r❡❛❧✐③❛❞❛ ♣♦r ❆rq✉✐♠❡❞❡s✱ s✉r❣❡ ♣r♦✈❛✲ ✈❡❧♠❡♥t❡ ❛ ♣r✐♠❡✐r❛ sér✐❡ ✐♥✜♥✐t❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ✉♠❛ P✳●✳ ❞❡ r❛③ã♦ 1

4.

▼♦str❛r❡♠♦s ❛ s❡❣✉✐r ♦ ♣r♦❝❡ss♦ ✉t✐❧✐③❛❞♦ ♣♦r ❆rq✉✐♠❡❞❡s ♣❛r❛ ❡♥❝♦♥tr❛r ❛ s♦♠❛ ❞❡ss❛ sér✐❡✱ ❡✈✐t❛♥❞♦ ❢❛③❡r n→ ∞✳

Pr♦❜❧❡♠❛✿ ▼♦str❛r q✉❡ 1+1 4+

1

42 +...+

1

4n +...= 4 3.

❙❡❣✉♥❞♦ ❆rq✉✐♠❡❞❡s✱ 1+1 4+

1

42 +...+

1 4n +

1 3·

1 4n =

4 3.

■ss♦ s❡❣✉❡ ❞♦ s❡❣✉✐♥t❡ ❢❛t♦✿ 1

4k + 1 3·

1 4k =

4 3.4k =

1 3 ·

1 4k−1

❆ss✐♠✱

1 + 1 4+

1

42 +...+

1 4n +

1 3. 1 4n =

= 1 + 1 4+

1

42 +...+

1 4n−1 +

1 3.

1 4n−1

=...=

= 1 +

1 4+ 1 3. 1 4

= 1 +1 3 =

4 3

✻ ❆ ❝❛rt❛ ❛ ❊r❛tóst❡♥❡s s♦❜r❡ ♦ ♠ét♦❞♦

❯♠ ♥♦✈♦ ❧✐✈r♦ ❞❡ ❆rq✉✐♠❡❞❡s ❢♦✐ ❞❡s❝♦❜❡rt♦ ❡♠ ✶✾✵✻✱ ❡♠ ❈♦♥st❛♥t✐♥♦♣❧❛✱ ♣❡❧♦ ✜❧ó❧♦❣♦ ❞✐♥❛♠❛rq✉ês ❏✳ ▲✳ ❍❡✐❜❡r❣ ✭✶✽✺✹ ✕ ✶✾✷✽✮✳ ❊st❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏❖ ▼ét♦❞♦✑✱ ❥✉st❛♠❡♥t❡ ♣♦rq✉❡ ♥❡❧❡ ♦ ❣❡ô♠❡tr❛ ❣r❡❣♦ ❞❡s❝r❡✈❡ ✉♠ ✏♠ét♦❞♦ ♠❡❝â♥✐❝♦✑ ♣❛r❛ ✐♥✈❡st✐❣❛r q✉❡stõ❡s ♠❛✲ t❡♠át✐❝❛s✳ ❆rq✉✐♠❡❞❡s t✐♥❤❛ ♦ ❝♦st✉♠❡ ❞❡ ❡♥✈✐❛r s✉❛s ♦❜r❛s ❛♦s sá❜✐♦s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ♣r❡❢❛❝✐❛♥❞♦✲❛s ❝♦♠ ❝❛rt❛s ❛ ❡ss❡s sá❜✐♦s✳ ❙❡✉ ❧✐✈r♦✱ ✏❖ ▼ét♦❞♦✑✱ ❝♦♥té♠ ❝♦♠♦ ♣r❡❢á❝✐♦ ✉♠❛ ❝❛rt❛ ❛ ❊r❛tóst❡♥❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛ q✉❛❧ ❝♦♠❡ç❛✈❛ ❛ss✐♠✿

(9)

❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

❆rq✉✐♠❡❞❡s ❛ ❊r❛tóst❡♥❡s✱ ❙❛✉❞❛çõ❡s

❊♥✈✐❡✐✲❧❤❡ ❡♠ ♦✉tr❛ ♦❝❛s✐ã♦ ❛❧❣✉♥s t❡♦r❡♠❛s ❞❡s❝♦❜❡rt♦s ♣♦r ♠✐♠✱ ♠❡r❛✲ ♠❡♥t❡ ♦s ❡♥✉♥❝✐❛❞♦s✱ ❞❡✐①❛♥❞♦✲❧❤❡ ❛ t❛r❡❢❛ ❞❡ ❞❡s❝♦❜r✐r ❛s ❞❡♠♦♥str❛çõ❡s ❡♥tã♦ ♦♠✐t✐❞❛s✳✳✳ ❱❡♥❞♦ ❡♠ ✈♦❝ê ✉♠ ❞❡❞✐❝❛❞♦ ❡st✉❞✐♦s♦✱ ❞❡ ❝♦♥s✐❞❡rá✈❡❧ ✐♠✐♥ê♥❝✐❛ ❡♠ ❋✐❧♦s♦✜❛ ❡ ✉♠ ❛❞♠✐r❛❞♦r ❞❛ ♣❡sq✉✐s❛ ♠❛t❡♠át✐❝❛✱ ❥✉❧❣✉❡✐ ❝♦♥✈❡♥✐❡♥t❡ ❡s❝r❡✈❡r✲❧❤❡ ♣❛r❛ ❡①♣❧✐❝❛r ❛s ♣❡❝✉❧✐❛r✐❞❛❞❡s ❞❡ ✉♠ ❝❡rt♦ ♠é✲ t♦❞♦ ♣❡❧♦ q✉❛❧ é ♣♦ssí✈❡❧ ✐♥✈❡st✐❣❛r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❞❡ ▼❛t❡♠át✐❝❛ ♣♦r ♠❡✐♦s ♠❡❝â♥✐❝♦s✳✳✳ ❈❡rt❛s ❝♦✐s❛s ♣r✐♠❡✐r♦ s❡ t♦r♥❛r❛♠ ❝❧❛r❛s ♣❛r❛ ♠✐♠ ♣❡❧♦ ♠ét♦❞♦ ♠❡❝â♥✐❝♦✱ ❡♠❜♦r❛ ❞❡♣♦✐s t✐✈❡ss❡♠ ❞❡ s❡r ❞❡♠♦♥str❛❞❛s ♣❡❧❛ ●❡♦♠❡tr✐❛✱ ❥á q✉❡ s✉❛ ✐♥✈❡st✐❣❛çã♦ ♣❡❧♦ r❡❢❡r✐❞♦ ♠ét♦❞♦ ♥ã♦ ❝♦♥❞✉③✐ss❡ ❛ ♣r♦✈❛s ❛❝❡✐tá✈❡✐s✳ ❈❡rt❛♠❡♥t❡ é ♠❛✐s ❢á❝✐❧ ❢❛③❡r ❛s ❞❡♠♦♥str❛çõ❡s q✉❛♥❞♦ t❡♠♦s ♣r❡✈✐❛♠❡♥t❡ ❛❞q✉✐r✐❞♦✱ ♣❡❧♦ ♠ét♦❞♦✱ ❛❧❣✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s q✉❡s✲ tõ❡s ❞♦ q✉❡ s❡♠ ❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦✳✳✳ ❊st♦✉ ❝♦♥✈❡♥❝✐❞♦ ❞❡ q✉❡ s❡rá ✈❛❧✐♦s♦ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛✱ ♣♦✐s ♣r❡ss✐♥t♦ q✉❡ ♦✉tr♦s ✐♥✈❡st✐❣❛❞♦r❡s ❞❛ ❛t✉❛❧✐❞❛❞❡ ♦✉ ❞♦ ❢✉t✉r♦ ❞❡s❝♦❜r✐rã♦✱ ♣❡❧♦ ♠ét♦❞♦ ❛q✉✐ ❞❡s❝r✐t♦✱ ♦✉tr❛s ♣r♦♣♦s✐çõ❡s q✉❡ ♥ã♦ ♠❡ ♦❝♦rr❡r❛♠✳

➱ ♦♣♦rt✉♥♦ ♥♦t❛r✱ ❛ ♣r♦♣ós✐t♦ ❞❛s ♣❛❧❛✈r❛s ✜♥❛✐s ❞❛ ❝✐t❛çã♦ ❛❝✐♠❛✱ q✉❡ ♦ ❝❤❛♠❛❞♦ ✏♠é✲ t♦❞♦ ❞♦s ✐♥❞✐✈✐sí✈❡✐s✑✱ ✐♥✈❡♥t❛❞♦ ♥♦ sé❝✉❧♦ ❳❱■■✱ ❡ q✉❡ ❞❡✉ ♦r✐❣❡♠ ❛♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ é ♠✉✐t♦ ♣❛r❡❝✐❞♦ ❝♦♠ ♦ ❛♥t✐❣♦ ✏♠ét♦❞♦ ♠❡❝â♥✐❝♦✑ ❞❡ ❆rq✉✐♠❡❞❡s✳ ❚❛♥t♦ ✉♠ q✉❛♥t♦ ♦✉tr♦ ❝❛r❡❝❡♠ ❞❡ ✉♠❛ ❢✉♥❞❛♠❡♥t❛çã♦ só❧✐❞❛✱ ♠❛s ❝♦♥tê♠ ♦s ✐♥❣r❡❞✐❡♥t❡s q✉❡ ❢❛❝✐❧✐t❛♠ ❛s ❞❡s❝♦❜❡rt❛s ❡ q✉❡✱ ♥♦ sé❝✉❧♦ ❳❱■■✱ ❢♦r❛♠ ❞❡❝✐s✐✈♦s ♣❛r❛ ❣r❛♥❞❡s ❛✈❛♥ç♦s ❞❛ ♠❛t❡♠át✐❝❛✳

✼ ❆ q✉❛❞r❛t✉r❛ ❞❛ ♣❛rá❜♦❧❛ ♣❡❧♦ ▼ét♦❞♦ ❞❛ ❆❧❛✈❛♥❝❛

❖ ♠ét♦❞♦ q✉❡ ❆rq✉✐♠❡❞❡s ✈✐s✉❛❧✐③♦✉ ❝♦rr❡t❛♠❡♥t❡ ❡ q✉❡ ❤❛❜✐❧✐t❛r✐❛ s❡✉s ❝♦♥t❡♠♣♦râ♥❡♦s ❡ s✉❝❡ss♦r❡s ❛ ❢❛③❡r ♥♦✈❛s ❞❡s❝♦❜❡rt❛s✱ ❝♦♥s✐st✐❛ ♥✉♠ ❡sq✉❡♠❛ ♣❛r❛ ❡q✉✐❧✐❜r❛r ❡♥tr❡ s✐ ♦s ✏❡❧❡♠❡♥t♦s✑ ❞❡ ✜❣✉r❛s ❣❡♦♠étr✐❝❛s✳

❖ ♣r✐♠❡✐r♦ t❡♦r❡♠❛ q✉❡ ❆rq✉✐♠❡❞❡s ❞❡s❝♦❜r✐✉ ♠❡❞✐❛♥t❡ ❛ ♦♣❡r❛çã♦ ❞❡ ❡q✉✐❧✐❜r❛r ❡❧❡✲ ♠❡♥t♦s ❢♦✐ ♦ ❝é❧❡❜r❡ r❡s✉❧t❛❞♦ ❞❡ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ ♣❛rá❜♦❧❛ é ✹✴✸ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ q✉❡ t❡♠ ❛ ♠❡s♠❛ ❜❛s❡ ❡ ❛❧t✉r❛✳ ❊❧❡ ❝❤❡❣♦✉ ❛ ✐ss♦ ❡q✉✐❧✐❜r❛♥❞♦ ❡♥tr❡ s✐ ♦s s❡❣♠❡♥t♦s q✉❡ ❢♦r♠❛♠ ♦ tr✐â♥❣✉❧♦ ❝♦♠ ♦s s❡❣♠❡♥t♦s q✉❡ ❢♦r♠❛♠ ♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦✳

❆♣ós s✉❛s ❞❡s❝♦❜❡rt❛s ♣♦r ✏♠ét♦❞♦ ❞❛ ❛❧❛✈❛♥❝❛✑✱ ❡❧❡ ✉s❛✈❛ ♦ ✏♠ét♦❞♦ ❞❡ ❡①❛✉stã♦✑ ♣❛r❛ ♣r♦✈á✲❧❛s✱ ❛❥✉st❛♥❞♦✲s❡ ❛ss✐♠ ❛♦s ♣❛❞rõ❡s ❞❡ r✐❣♦r ❞❛ é♣♦❝❛✳

❙❡❥❛ s ❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣♦r ✉♠❛ ♣❛rá❜♦❧❛ p❡ ✉♠❛ ❝♦r❞❛ AB ❞❡ ♣♦♥t♦ ♠é❞✐♦ M✳ ❙❡❥❛ t

❛ t❛♥❣❡♥t❡ ❛p ❡♠ A✳

(10)

❉♦s ♣♦♥t♦s B ❡ M tr❛ç❛♠ s❡ r❡t❛s ♣❛r❛❧❡❧❛s ❛♦ ❡✐①♦✱ ❛s q✉❛✐s ✐♥t❡r❝❡♣t❛♠ t ❡♠ D ❡ E✱

r❡s♣❡❝t✐✈❛♠❡♥t❡❀ s✉♣♦♥❤❛♠♦s q✉❡M E ✐♥t❡r❝❡♣t❡p❡♠ C✱ ♣♦♥t♦ ❡st❡ ❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡ ❞❡ s✳ P♦r ✉♠ t❡♦r❡♠❛ ❛♥t❡r✐♦r ❝♦♥❤❡❝✐❞♦✱C é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ M E✳ ❙❡❥❛ l ❛ r❡t❛ q✉❡ ❝♦♥té♠ AC ❡ ✐♥❞✐q✉❡♠♦s ♣♦r F s✉❛ ✐♥t❡rs❡❝çã♦ ❝♦♠BD✳

◆❡st❛ ❛❧t✉r❛ ❆rq✉✐♠❡❞❡s ❝♦♠♣❛r❛ ♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ s ❝♦♠ ♦ tr✐â♥❣✉❧♦ ABD✳ ❙❡❥❛

❖ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡AB✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ r❡t❛ ♣♦rO✱ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❡p✐♥t❡r❝❡♣t❡ p✱ t ❡ l ♥♦s ♣♦♥t♦s P✱ Q ❡R✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✈✐❞♦ ❛ ♦✉tr♦ t❡♦r❡♠❛ ❝♦♥❤❡❝✐❞♦

OP OQ =

OB AB =

RF AF

◆❡st❡ ♣♦♥t♦ ❡❧❡ ❞á ✉♠ ♣❛ss♦ ❡♥❣❡♥❤♦s♦✿ ❝♦♥s✐❞❡r❛ l ❝♦♠♦ ✉♠❛ ❛❧❛✈❛♥❝❛✱ ❝♦♠ ❢✉❧❝r♦ ❡♠ F✱ ❡ t♦♠❛ ♦ ♣♦♥t♦T ❡♠ l ❞❡ ♠❛♥❡✐r❛ q✉❡ F s❡❥❛ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡AT✳ ❊♠ T ❡❧❡ ✏♣❡♥❞✉r❛✑

✉♠ s❡❣♠❡♥t♦U V✱ ❝♦♥❣r✉❡♥t❡ ❛ OP✳ ❊♥tã♦✱ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✿

U V OQ =

RF AF =

RF

T F, ou U V.T F =OQ.RF

❆ss✐♠ ♦ s❡❣♠❡♥t♦ U V✱ s✉s♣❡♥s♦ ♣❡❧♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦ T✱ ❡stá ❡♠ ❡q✉✐❧í❜r✐♦ ❝♦♠ ♦ s❡❣✲

♠❡♥t♦ OQ✱ s✉s♣❡♥s♦ ♣❡❧♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦ R✳ ❆rq✉✐♠❡❞❡s ✐♠❛❣✐♥❛ ❛❣♦r❛ ♦ tr✐â♥❣✉❧♦ABD

❝♦♠♦ ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s s❡❣♠❡♥t♦s ❝♦♠♦ OQ✱ ♣❛r❛❧❡❧♦s ❛♦ ❡✐①♦✳ ❈❛❞❛ ✉♠ ❞❡❧❡s t❡♠ ✉♠

s❡❣♠❡♥t♦ ❝♦rr❡s♣♦♥❞❡♥t❡OP ❝♦♥❣r✉❡♥t❡ ❛ ✉♠ s❡❣♠❡♥t♦U V✱ q✉❡ s❡ ✏♣❡♥❞✉r❛✑ ❡♠T✳ ❉❡st❛

❢♦r♠❛ ❡❧❡ ❝♦♥❝❡❜❡ ♦ tr✐â♥❣✉❧♦ ❡♠ ❡q✉✐❧í❜r✐♦ ❝♦♠ ♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ s✱ q✉❡ s❡ ✐♠❛❣✐♥❛

s✉s♣❡♥s♦ ❡♠ T✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ❝♦♠♦ s❡ s❛❜✐❛ ♣r❡✈✐❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r ♦ tr✐â♥❣✉❧♦

s✉s♣❡♥s♦ ♣❡❧♦ s❡✉ ❜❛r✐❝❡♥tr♦✱ q✉❡ é ♦ ♣♦♥t♦ G ❞❡ l t❛❧ q✉❡ F G = 1 3F A =

1

3F T✳ P♦rt❛♥t♦✱

s ❡ ♦ tr✐â♥❣✉❧♦ ABD ❚ê♠ ár❡❛s ❝✉❥❛ r❛③ã♦ é 1 : 3✳ ❋✐♥❛❧♠❡♥t❡✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABD

é ♦ q✉á❞r✉♣❧♦ ❞❡ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC✱ ❡ t❡♠♦s ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ❆rq✉✐♠❡❞❡s✿ ❛ ár❡❛ ❞♦

s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ é 4

3 ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❝♦♠ ❛ ♠❡s♠❛ ❜❛s❡ ❡ ♠❡s♠♦ ✈ért✐❝❡✳

✽ ❖ ❝♦♥t❛❞♦r ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛

❚❡♠♦s ❛q✉✐ ✉♠❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❆rq✉✐♠❡❞❡s à ❧♦❣íst✐❝❛ ✭❛r✐t♠ét✐❝❛ ❛♣❧✐❝❛❞❛✮✳

(11)

❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

❊❧❡ s❡ ❣❛❜❛✈❛ ❞❡ ♣♦❞❡r ❡s❝r❡✈❡r ✉♠ ♥ú♠❡r♦ ♠❛✐♦r ❞♦ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛ ♥❡❝❡ssár✐♦s ♣❛r❛ ❡♥❝❤❡r ♦ ✉♥✐✈❡rs♦✳

❈♦♠♦ q✉❛s❡ t♦❞♦s ♦s ❛strô♥♦♠♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❆rq✉✐♠❡❞❡s✱ ❝♦♥❝❡❜✐❛ ♦ ❯♥✐✈❡rs♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❡♥♦r♠❡ ❡s❢❡r❛✱ ❝♦♠ ❝❡♥tr♦ ♥❛ ❚❡rr❛ ✭✐♠ó✈❡❧✮ ❡ r❛✐♦ ✐❣✉❛❧ à ❞✐stâ♥❝✐❛ ❞❛ ❚❡rr❛ ❛♦ ❙♦❧✳

❙✉❜❡st✐♠❛♥❞♦ ♦ t❛♠❛♥❤♦ ❞❡ ✉♠ ❣rã♦ ❞❡ ❛r❡✐❛✱ ❆rq✉✐♠❡❞❡s ❛❞♠✐t✐✉ q✉❡ 10.000 ❞❡ss❡s

❣rã♦s ♣r❡❡♥❝❤❡ss❡♠ ♦ ❡s♣❛ç♦ ♦❝✉♣❛❞♦ ♣♦r ✉♠❛ s❡♠❡♥t❡ ❞❡ ♣❛♣♦✉❧❛❀ ❡ q✉❡40❞❡ss❛s s❡♠❡♥t❡s✱

❥✉st❛♣♦st❛s ❧❛❞♦ ❛ ❧❛❞♦✱ ❡①❝❡❞❡r✐❛♠ ❛ ❧❛r❣✉r❛ ❞❡ ✉♠ ❞❡❞♦✳ ❉❛í ❝♦♥❝❧✉✐✉ ✭✉s❛♥❞♦ ❛ r❡❧❛çã♦

V = πd3

6 < d 3

, ♦♥❞❡ ❞ é ♦ ❞✐â♠❡tr♦ ❡ V é ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✮ q✉❡ ✉♠❛ ❡s❢❡r❛ ❞❡

❞✐â♠❡tr♦ ✐❣✉❛❧ à ❧❛r❣✉r❛ ❞❡ ✉♠ ❞❡❞♦ ♥ã♦ ❝♦♥té♠ ♠❛✐s q✉❡403

= 64000s❡♠❡♥t❡s ❞❡ ♣❛♣♦✉❧❛

❡✱ ♣♦rt❛♥t♦✱ ♥❡❧❛ ♥ã♦ ❝❛❜❡♠ ♠❛✐s q✉❡ 10.000× 64.000 = 640 ♠✐❧❤õ❡s ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛✱

s❡❣✉r❛♠❡♥t❡✱ ❡♥tã♦✱ ♥❡ss❛ ❡s❢❡r❛ ❝♦♠♣♦rt❛ ♠❡♥♦s ❞❡1❜✐❧❤ã♦✱ ✐st♦ é✱109

❞❡ ❣rã♦s ❞❡ ❛r❡✐❛✳ ❆ s❡❣✉✐r ❆rq✉✐♠❡❞❡s ✐♥tr♦❞✉③ ❡♠ s❡✉ r❛❝✐♦❝í♥✐♦ ♦ ❡stá❞✐♦ ✭✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡q✉✐✈❛❧❡♥t❡ ❛ ❝❡r❝❛ ❞❡ 160m✮ q✉❡ ❡st✐♠♦✉ ❡♠ ♠❡♥♦s ❞❡ 104 ❧❛r❣✉r❛s ❞❡ ❞❡❞♦s✳ ❈♦♠♦ ♦s

✈♦❧✉♠❡s ❞❡ ❞✉❛s ❡s❢❡r❛s ❡stã♦ ❡♥tr❡ s✐ ♥❛ r❛③ã♦ ❞♦s ❝✉❜♦s ❞❡ s❡✉s ❞✐â♠❡tr♦s✱ ♦ ♥ú♠❡r♦ ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛ ♥❡❝❡ssár✐♦ ♣❛r❛ ♣r❡❡♥❝❤❡r ✉♠❛ ❡s❢❡r❛ ❞❡ ❞✐â♠❡tr♦ ✐❣✉❛❧ ❛ ✉♠ ❡stá❞✐♦ é ♠❡♥♦r q✉❡109

.(104

)3 = 1021✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ❞❛❞♦s ❞❡ ♠❡❞✐❞❛s ❛str♦♥ô♠✐❝❛s ❝♦♥❤❡❝✐❞❛s ❡♠ s✉❛ é♣♦❝❛✱ ♥ã♦ ❧❤❡ ❢♦✐ ❞✐❢í❝✐❧ ❡st❛❜❡❧❡❝❡r q✉❡ ♦ ❞✐â♠❡tr♦ ❞♦ ✉♥✐✈❡rs♦ ❡r❛ ✐♥❢❡r✐♦r ❛1010 ❡stá❞✐♦s✳

❊♥tã♦ r❡♣❡t✐♥❞♦ ❛ ❛r❣✉♠❡♥t❛çã♦ ❛♥t❡r✐♦r✱ ❝♦♥❝❧✉✐✉ q✉❡ ♣❛r❛ ♣r❡❡♥❝❤❡r t♦t❛❧♠❡♥t❡ ❡st❡ ✉♥✐✈❡rs♦ ❜❛st❛r✐❛ ✉♠ ♥ú♠❡r♦ ❞❡ ❣rã♦s ✐♥❢❡r✐♦r ❛1021

.(1010

)3 = 1051

.

❈♦♥t❛r ❣rã♦s ❞❡ ❛r❡✐❛ ♣♦❞❡ t❡r s✐❞♦ ♣❛r❛ ❆rq✉✐♠❡❞❡s ❛♣❡♥❛s ✉♠ ❡①❡r❝í❝✐♦ ♣❛r❛ ♣♦r ❡♠ ♣rát✐❝❛ ✉♠ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦✱ q✉❡ ❝r✐♦✉✱ ❞❡ ❜❛s❡ 108

, ♣❛r❛ ❡①♣r✐♠✐r ♥ú♠❡r♦s ♠✉✐t♦

❣r❛♥❞❡s✱ ❥á q✉❡ ♦ s✐st❡♠❛ ❛❧❢❛❜ét✐❝♦ ❡♠ ✉s♦ ♥❛ ●ré❝✐❛ ❡r❛ ❞❡✜❝✐❡♥t❡ q✉❛♥t♦ ❛ ❡st❡ ❛s♣❡❝t♦✳

✾ ❙♦❜r❡ ♦s ❈♦r♣♦s ❋❧✉t✉❛♥t❡s ✭❆ ❈♦r♦❛ ❞♦ ❘❡✐✮

❱❡r❡♠♦s ❝♦♠♦ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦r♦❛ ✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❆rq✉✐♠❡❞❡s ❡ ✉♠ ♣♦✉❝♦ ❞❡ ♣r♦♣♦rçõ❡s✳ ❙❡❥❛P ♦ ♣❡s♦ ❞❛ ❝♦r♦❛✱ q✉❡ s✉♣♦♠♦s t❡r s✐❞♦ ❢❡✐t❛ ❝♦♠ ✉♠ ♣❡s♦ x❞❡

♦✉r♦ ❡ ✉♠ ♣❡s♦y ❞❡ ♣r❛t❛✳ ▲♦❣♦✿

P =x+y (1)

❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠❛ ♣♦rçã♦ ❞❡ ♦✉r♦ ❞❡ ♣❡s♦ x t❡♥❤❛ ♣❡s♦ x✬ q✉❛♥❞♦ ♣❡s❛❞❛ ❞❡♥tr♦

❞✬á❣✉❛✱ ❡ s❡❥❛ X✬ ♦ ♣❡s♦✱ ❞❡♥tr♦ ❞✬á❣✉❛✱ ❞❡ ✉♠❛ ♣♦rçã♦ ❞❡ ♦✉r♦ ❞❡ ♣❡s♦ ✐❣✉❛❧ ❛♦ ♣❡s♦ P ❞❛

❝♦r♦❛✳ ❖r❛✱ ♦ ♣❡s♦ ❞♦ ♦✉r♦ ❞❡♥tr♦ ❞✬á❣✉❛ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ s❡✉ ♣❡s♦ ❢♦r❛ ❞✬á❣✉❛ ✭♣♦rq✉❡ ♦ ✈♦❧✉♠❡ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♣❡s♦✱ ❞❡✈✐❞♦ à ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❞♦ ♠❛t❡r✐❛❧✮✳ ▲♦❣♦✱

x′

x = X′

P − x

′ ₌ xX′

P (2)

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♦ ♣❡s♦ ❞❛ ♣r❛t❛✱ q✉❛♥❞♦ ♣❡s❛❞❛ ❞❡♥tr♦ ❞✬á❣✉❛✱ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ s❡✉ ♣❡s♦ ❢♦r❛ ❞✬á❣✉❛✳ ❙❡ y✬ ❞❡s✐❣♥❛ ♦ ♣❡s♦✱ ❞❡♥tr♦ ❞✬á❣✉❛✱ ❞❡ ✉♠❛ ♣♦rçã♦ ❞❡ ♣r❛t❛ ❞❡ ♣❡s♦ y✱ ❡

(12)

Y✬ ♦ ♣❡s♦✱ ❞❡♥tr♦ ❞✬á❣✉❛✱ ❞❡ ✉♠❛ ♣♦rçã♦ ❞❡ ♣r❛t❛ ❞❡ ♣❡s♦ ✐❣✉❛❧ ❛♦ ♣❡s♦ P ❞❛ ❝♦r♦❛✱ ❡♥tã♦

t❡r❡♠♦s✱ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ♥♦ r❛❝✐♦❝í♥✐♦ q✉❡ ♥♦s ❧❡✈♦✉ á ❡q✉❛çã♦(2) ❛❝✐♠❛✱

y′ ₌ yY′

P (3)

❙❡❥❛P✬ ♦ ♣❡s♦ ❞❛ ❝♦r♦❛ q✉❛♥❞♦ ♣❡s❛❞❛ ❞❡♥tr♦ ❞✬á❣✉❛✳ ➱ ❝❧❛r♦ q✉❡P✬=x✬+y✬✱ ❞❡ s♦rt❡

q✉❡✱ s♦♠❛♥❞♦(2) ❡(3) ❛❝✐♠❛✱ ♦❜t❡♠♦s

P′ ₌_x′₊_y′ ₌ xX′+yY′

P ∴ P P

′ ₌_xX′₊_yY′

❉❛q✉✐ ❡ ❞❡ (1) s❡❣✉❡✲s❡ q✉❡

(x+y)P′ ₌_xX′₊_yY′ _∴ _x₍_X′₋_P′_{) =}_y₍_P′₋_Y′₎_,

♦✉ ❛✐♥❞❛✱

x y =

P′₋_Y′

X′₋_P′ (4)

◆ã♦ t❡♠♦s ❞❛❞♦s ❡s♣❡❝í✜❝♦s s♦❜r❡ ❛ ❝♦r♦❛ ✈❡r❞❛❞❡✐r❛ q✉❡ ♦ r❡✐ ❍✐❡rã♦ ❡♥tr❡❣♦✉ ❛ ❆r✲ q✉✐♠❡❞❡s ♣❛r❛ s❡r ✐♥✈❡st✐❣❛❞❛ ♠❛s ♣♦❞❡♠♦s ♠✉✐t♦ ❜❡♠ ✐♠❛❣✐♥❛r ✉♠❛ s✐t✉❛çã♦ ❝♦♥❝r❡t❛✳ ❉✐❣❛♠♦s q✉❡ ❛ ❝♦r♦❛ ♣❡s❛ss❡ P = 894g ❢♦r❛ ❞✬á❣✉❛ ❡ 834g ❞❡♥tr♦ ❞✬á❣✉❛✳ ❙✉♣♦♥❤❛♠♦s

t❛♠❜é♠✱ s❡❣✉✐♥❞♦ ❛ ♥♦t❛çã♦ ❥á ✐♥tr♦❞✉③✐❞❛✱ q✉❡ X✬ = 847,7g ❡ Y✬ = 809g✳ ❙✉❜st✐t✉✐♥❞♦

❡st❡s ✈❛❧♦r❡s ❡♠(4) ❡♥❝♦♥tr❛♠♦s

x y =

834−809 847,7−834 =

25 13,7

∼

= 1,82

❉❛q✉✐ ❡ ❞❡ (1) ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♣❛r❛ ❞❡t❡r♠✐♥❛r x ❡y :

x+y= 894, x= 1,82y

❘❡s♦❧✈❡♥❞♦ ❡st❡ s✐st❡♠❛ ❡♥❝♦♥tr❛♠♦s x ∼= 577g ❡ y ∼= 317g✳ ♣♦rt❛♥t♦✱ ♥♦ss❛ ❝♦r♦❛

✐♠❛❣✐♥ár✐❛ ❝♦♥té♠577g ❞❡ ♦✉r♦ ❡ 317g ❞❡ ♣r❛t❛✳

❚❡♥❞♦ ❡♠ ❝♦♥t❛ q✉❡ ♦ ♣❡s♦ ❡s♣❡❝í✜❝♦ ❞♦ ♦✉r♦ é 19,3g/cm3

❡ ♦ ❞❛ ♣r❛t❛ é 10,5g/cm3

✱ ♣♦❞❡♠♦s ♣r♦ss❡❣✉✐r ❡ ❝❛❧❝✉❧❛r ❛s q✉❛♥t✐❞❛❞❡s ✈♦❧✉♠étr✐❝❛s ❞❡ ♦✉r♦ ❡ ♣r❛t❛ ✉s❛❞♦s ♥❛ ❝♦r♦❛✳ ❚r❛t❛✲s❡✱ ♥♦✈❛♠❡♥t❡✱ ❞❡ ✉♠ ❝á❧❝✉❧♦ s✐♠♣❧❡s ✉s❛♥❞♦ ♣r♦♣♦rçõ❡s✳ ❙❡❥❛♠ V0 ❡ V_p✱ r❡s♣❡❝t✐✈❛✲

♠❡♥t❡✱ ♦s ✈♦❧✉♠❡s ❞❡ ♦✉r♦ ❡ ♣r❛t❛ ❡♠♣r❡❣❛❞♦s ♣❛r❛ ❢❛③❡r ❛ ❝♦r♦❛✳ ❊♥tã♦✱

x V0

= 19,3 1 e

y Vp

= 10,5 1

❙✉❜st✐t✉✐♥❞♦x= 577 ❡ y= 317 ❡ r❡s♦❧✈❡♥❞♦ ❛s ❡q✉❛çõ❡s r❡s✉❧t❛♥t❡s ❡♥❝♦♥tr❛♠♦s

V0 =

577 19,3

∼

= 29,9cm3

e Vp = 317 10,5

∼

= 30,2cm3

(13)

❘❊▼❛t

■❙❙◆ ✷✶✼✼✲✺✵✾✺

♥♦_{✸ ✲ ✷✵✶✸}

❘❊❱■❙❚❆ ❊▲❊❚❘Ô◆■❈❆ ❉❊ ▼❆❚❊▼➪❚■❈❆

✇✇✇✷✳❥❛t❛✐✳✉❢❣✳❜r✴♦❥s✴✐♥❞❡①✳♣❤♣✴♠❛t❡♠❛t✐❝❛ ❝♦♥t❛t♦✿ r❡♠❛t✳✉❢❣❅❣♠❛✐❧✳❝♦♠

❱❡♠♦s q✉❡ ♦ ♦✉r✐✈❡s ✉s♦✉ ♣r❛t✐❝❛♠❡♥t❡ ❛s ♠❡s♠❛s q✉❛♥t✐❞❛❞❡s ✈♦❧✉♠étr✐❝❛s ❞❡ ♦✉r♦ ❡ ♣r❛t❛✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 30cm3 ❞❡ ♦✉r♦ ❡

30cm3 ❞❡ ♣r❛t❛✳ ➱ ♠✉✐t❛ ♣r❛t❛ ♣❛r❛ ♣♦✉❝♦ ♦✉r♦

♥✉♠❛ ❝♦r♦❛ r❡❛❧✦ ❖①❛❧á ✐st♦ ♥ã♦ t❡♥❤❛ ❝✉st❛❞♦ ❛ ❝❛❜❡ç❛ ❞♦ ♦✉r✐✈❡s✳✳✳

❘❡❢❡rê♥❝✐❛s

❬❆❆❇❖❊✲✶✾✽✹❪ ❆❆❇❖❊✱ ❆✳ ❊♣✐só❞✐♦s ❞❛ ❤✐stór✐❛ ❛♥t✐❣❛ ❞❛ ♠❛t❡♠át✐❝❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ✶✾✽✹✳

❬➪❱■▲❆✲✶✾✽✼❪ ➪❱■▲❆✱ ●✳ ❆rq✉✐♠❡❞❡s✱ ❆ ❡s❢❡r❛ ❡ ♦ ❝✐❧✐♥❞r♦✳ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡✲ ♠át✐❝❛ ✶✵✱ ✶✶✲✷✵✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ✶✾✽✼✳ ❬➪❱■▲❆✲✶✾✽✻❪ ❆rq✉✐♠❡❞❡s✱ ♦ r✐❣♦r ❡ ♦ ♠ét♦❞♦✳ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ✹✱ ✷✼✲✹✺✱

❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ✶✾✽✻✳

❬❇❖❨❊❘✲✶✾✾✻❪ ❇❖❨❊❘✱ ❈✳ ❇✳ ❍✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ✭✷❛ ❊❞✐çã♦✮✳ ❙ã♦ P❛✉❧♦✿ ❊❞❣❛r❞ ❇❧ü❝❤❡r✱ ✶✾✾✻✳

❬❇❖❨❊❘✲✶✾✾✷❪ ❈á❧❝✉❧♦✳ ❙ã♦ P❛✉❧♦✿ ❆t✉❛❧✱ ✶✾✾✷✳

❬❙❖❯❩❆✲✶✾✽✻❪ ❙❖❯❩❆✱ ❙✳ ❆rq✉✐♠❡❞❡s ❡ ❛ ❝♦r♦❛ ❞♦ r❡✐✿ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ✾✱ ✶✶✲✶✺✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ✶✾✽✻✳