2011 Λύσεις Σχ. β&I

122 109800000 îùôèîôè 7 ó π ó âóè ë

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1,02 10 ≈ 90 . 1,22 109800000 οÙÔÈÎÔÈ. 7. ∞Ó π Ó Â›Ó·È Ë ¤ÓÙ·ÛË ÙÔ˘ ʈÙfi˜ ·ÊÔ‡ ‰È¤ÏıÂÈ Ì¤Û· ·fi Ó Ê›ÏÙÚ·, ÙfiÙÂ Ë ¤ÓÙ·Û‹ ÙÔ˘ ·ÊÔ‡ ‰È¤ÏıÂÈ Î·È Ì¤Û· ·fi ÙÔ ÂfiÌÂÓÔ Ê›ÏÙÚÔ, ‰ËÏ·‰‹ ·ÊÔ‡ ‰È¤ÏıÂÈ Û˘ÓÔÏÈο ̤۷ ·fi Ó + 1 Ê›ÏÙÚ· ı· Â›Ó·È π Ó + 1 = π Ó π Ó = 0,9π Ó . ÕÚ· Ô ·Ó·‰ÚÔÌÈÎfi˜ Ù‡Ô˜ Ù˘ ·ÎÔÏÔ˘ı›·˜ Â›Ó·È π Ó + 1 = 0,9 . π Ó . 10 100 2 100 2 10 – 1 2 – 1 3 3 3 3 3 3 3 3 3 3 . 3 3 3 5.3. °ÂˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ 75 {
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∂Âȉ‹ π 1 = π 0 . 0,9 Î·È π Ó + 1 = 0,9 . π Ó Ë ·ÎÔÏÔ˘ı›· Â›Ó·È ÁˆÌÂÙÚÈ΋ ÚfiÔ- ‰Ô˜ Ì 1Ô fiÚÔ π 0 . 0,9 Î·È ÏfiÁÔ Ï = 0,9, ¿Ú· π Ó = π 0 . 0,9 . 0,9 Ó – 1 π Ó = π 0 . 0,9 Ó . °È· Ó = 10 ¤¯Ô˘Ì π 10 = π 0 . 0,9 10 ≈ 0,35 . π 0 . 8. i) √È 11 ÂӉȿÌÂÛÔÈ ÙfiÓÔÈ Ì ÙÔ˘˜ ‰‡Ô ·ÎÚ·›Ô˘˜ Cã Î·È Cãã ı· Û¯ËÌ·Ù›˙Ô˘Ó ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô Ì · 1 = 261 Î·È · 13 = 522. ∂Âȉ‹ · 13 = · 1 . Ï 12 ¤¯Ô˘Ì 522 = 261 . Ï 12 Î·È ÂÔ̤ӈ˜ ii) ∏ Û˘¯ÓfiÙËÙ· ÙÔ˘ 5Ô˘ ÙfiÓÔ˘ ı· Â›Ó·È · 5 = · 1 . Ï 5 = 261 . 9. i) ∞Ó D Ó Â›Ó·È Ë ÔÛfiÙËÙ· ÙÔ˘ ÓÂÚÔ‡ ÛÙÔ „˘Á›Ô, ·ÊÔ‡ ÂÊ·ÚÌfiÛÔ˘Ì ÙË ‰È·‰Èηۛ· Ó ÊÔÚ¤˜, ÙfiÙÂ Ë ÔÛfiÙËÙ· ÙÔ˘ ÓÂÚÔ‡ ÛÙÔ „˘Á›Ô, ·Ó ÂÊ·ÚÌfi- ÛÔ˘Ì ÙË ‰È·‰Èηۛ· ÌÈ· ·ÎfiÌ· ÊÔÚ¿, ‰ËÏ·‰‹ Ó + 1 Û˘ÓÔÏÈο ÊÔÚ¤˜ ı· Â›Ó·È D Ó + 1 = D Ó . 4 = D Ó – 0,1 . D Ó = (1 – 0,1)D Ó = 0,9D Ó . ∂Ô̤ӈ˜ D Ó + 1 = 0,9D Ó Î·È D 1 = 36 fiÛÔ ÙÔ ÓÂÚfi Ô˘ ̤ÓÂÈ ÙËÓ 1Ë ÊÔÚ¿. μϤÔ˘Ì fiÙÈ Ë ·ÎÔÏÔ˘ı›· D Ó Â›Ó·È ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ Ì D 1 = 36 Î·È ÏfiÁÔ Ï = 0,9, ¿Ú· D Ó = 36 . 0,9 Ó –1 . ii) D 7 = 36 . 0,9 6 ≈ 19,13, ÔfiÙÂ Ë ÔÛfiÙËÙ· ÙÔ˘ ·ÓÙÈ˘ÎÙÈÎÔ‡ Â›Ó·È Â- Ú›Ô˘ 40 – 19,13 = 20,87 . 10. ∞ÊÔ‡ ‰ÈÏ·ÛÈ¿˙Ô˘Ì οı ÊÔÚ¿ ÙÔÓ Ú˘ıÌfi ÙˆÓ ÎfiÎÎˆÓ ÙÔ˘ Ú˘˙ÈÔ‡ ¤¯Ô˘- Ì · Ó + 1 = 2 . · Ó . ∂Âȉ‹ ÛÙÔ 1Ô ÙÂÙÚ·ÁˆÓ¿ÎÈ ‚¿˙Ô˘Ì 1 ÎfiÎÎÔ Ú‡˙È ¤¯Ô˘Ì · 1 = 1. ∂Ô̤ӈ˜ Ë ·ÎÔÏÔ˘ı›· ·Ó, Â›Ó·È ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ Ì · 1 = 1 Î·È ÏfiÁÔ Ï = 2, ¿Ú· · Ó = 1 . 2 Ó – 1 · Ó = 2 Ó – 1 . ™˘ÓÔÏÈο Û fiÏ· Ù· ÙÂÙÚ·ÁˆÓ¿ÎÈ· Ú¤ÂÈ Ó· ÌÔ˘Ó S 64 = 1 . = 2 64 – 1 ÎfiÎÎÔÈ Ú‡˙È. ΔÔ Ú‡˙È ·˘Ùfi Â›Ó·È ÂÚ›Ô˘ Û ÎÈÏ¿ 11. i) Œ¯Ô˘Ì S 1 = 3 S 2 = 3 .
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