2011 Λύσεις Σχ. β&I

Âè ùô ïùô ùë á êú âóè 8m ùô ú

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∂Âȉ‹ ÙÔ Ï¿ÙÔ˜ Ù˘ Á¤- Ê˘Ú·˜ Â›Ó·È 8m, ÙÔ ·Ú·- ‚ÔÏÈÎfi ÙfiÍÔ ı· Ù¤ÌÓÂÈ ÙÔÓ ¿ÍÔÓ· xãx ÛÙ· ÛËÌ›· μ(4, 0) Î·È μã(–4, 0), ÙˆÓ ÔÔ›ˆÓ ÔÈ Û˘ÓÙÂÙ·Á̤Ó˜ ı· Â·ÏËıÂ‡Ô˘Ó ÙËÓ Â͛ۈÛË (1). ∂Ô̤ӈ˜ ı· ÈÛ¯‡ÂÈ 0 = ·4 2 + 5,6 · = –0,35. ÕÚ·, ÙÔ ·Ú·‚ÔÏÈÎfi ÙfiÍÔ ¤¯ÂÈ Â͛ۈÛË y = –0,35x 2 + 5,6 Ì –4≤ x ≤ 4 (2) ∂Âȉ‹ ÙÔ ‡„Ô˜ Ù˘ ηÚfiÙÛ·˜ Â›Ó·È 2m Ù¤ÌÓÂÈ ÙÔ ·Ú·‚ÔÏÈÎfi ÙfiÍÔ ÛÙ· ÛË- Ì›· ∞ Î·È ∞ã, ÁÈ· Ó· ÂÚ¿ÛÂÈ ÙÔ ÁˆÚÁÈÎfi Ì˯¿ÓËÌ· ı· Ú¤ÂÈ ∞∞ã > 6m, Ô˘ Â›Ó·È ÙÔ Ï¿ÙÔ˜ ÙÔ˘ ÊÔÚÙËÁÔ‡. °È· Ó· ‚Úԇ̠ÙÔ ∞∞ã ·ÚΛ ·Ó ‚Úԇ̠ÙȘ Û˘ÓÙÂÙ·Á̤Ó˜ ÙˆÓ ∞, ∞ã. ∞Ó ı¤ÛÔ˘Ì ÛÙËÓ Â͛ۈÛË (2) y = 2 ‚Ú›ÛÎÔ˘Ì –0,35x 2 + 5,6 = 2 x 2 ≈ 10,6 x ≈ 3,2. ÕÚ· ∞(3,2, 0) Î·È ∞ã(–3,2, 0), ÔfiÙ ∞∞ã ≈ 6,4m > 6m. ∂Ô̤ӈ˜ ÙÔ ÁˆÚ- ÁÈÎfi Ì˯¿ÓËÌ· ÌÔÚ› Ó· ÂÚ¿ÛÂÈ. 16. i) ¢È·ÎÚ›ÓÔ˘Ì ÙÚÂȘ ÂÚÈÙÒÛÂȘ ñ ŸÙ·Ó ÙÔ ÛËÌÂ›Ô ª ‰È·ÁÚ¿ÊÂÈ ÙÔ Â˘ı. ÙÌ‹Ì· ∞μ, ‰ËÏ·‰‹ fiÙ·Ó 0 ≤ x ≤ 20, ÙfiÙ ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ÛÎÈ·Ṳ̂ÓÔ˘ ¯ˆÚ›Ô˘ √∞ª ı· Â›Ó·È ›ÛÔ Ì ofiÙ ı· Â›Ó·È f(x) = 5x. ñ ŸÙ·Ó ÙÔ ÛËÌÂ›Ô ª ‰È·ÁÚ¿ÊÂÈ ÙÔ Â˘ı. ÙÌ‹Ì· μ°, ‰ËÏ·‰‹ fiÙ·Ó 20 ≤ x ≤ 40, ÙfiÙ ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ÛÎÈ·Ṳ̂ÓÔ˘ ¯ˆÚ›Ô˘ ı· Â›Ó·È ›ÛÔ Ì ofiÙ ı· Â›Ó·È f(x) = 10x – 100. ñ ŸÙ·Ó ÙÔ ÛËÌÂ›Ô ª ‰È·ÁÚ¿ÊÂÈ ÙÔ Â˘ı. ÙÌ‹Ì· °¢, ‰ËÏ·‰‹ fiÙ·Ó 40 ≤ x ≤ 60, ÙfiÙ ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ÛÎÈ·Ṳ̂ÓÔ˘ ¯ˆÚ›Ô˘ ı· Â›Ó·È ›ÛÔ Ì ofiÙ ı· Â›Ó·È f(x) = 5x + 100. ∂ = ∂ ∞μ°¢ – ∂ √¢ª = 20 20 – 10 (60 – x) 2 = 5x + 100 ∂ = √∞ + μª 2 ∞μ = 10 + (x – 20) 2 20 = 10(x – 10) ∂ = √∞ ∞ª 2 = 10 x 2 = 5x A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 114
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∂Ô̤ӈ˜, Â›Ó·È 5x, 0 ≤ x ≤ 20 f(x) = 10x – 100, 20 ≤ x ≤ 40 5x + 100, 40 ≤ x ≤ 60. ii) ∏ ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f Â›Ó·È Ë ÔÏ˘ÁˆÓÈ΋ ÁÚ·ÌÌ‹ ÙÔ˘ ·Ú·Î¿- Ùˆ Û¯‹Ì·ÙÔ˜. iii) ∞fi ÙËÓ ·Ú·¿Óˆ ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË ÚÔ·ÙÂÈ fiÙÈ Ë f ·›ÚÓÂÈ ÙËÓ ÙÈÌ‹ 120, fiÙ·Ó x ÌÂٷ͇ 20 Î·È 40. ∂Ô̤ӈ˜ f(x) = 120 10x – 100 = 120 x = 22. 17. i) ∂›Ó·È ∂Ô̤ӈ˜ f(x) = x, 0 ≤ x ≤ 2 Î·È g(x) = –0,5x 2 + 2, 0 ≤ x ≤ 2 ª™°¢ = ª™ + °¢ 2 ™¢ = x + 2 2 (2 – x) = 4 – x 2 2 = –0,5x 2 + 2 ª∞μ = ∞μ ªƒ 2 = ∞μ ∞ƒ 2 = 2 x 2 = x Î·È A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 115 {
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ii) ∂›Ó·È
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