This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hili ni swali la kwanza ulilotuma, na sasa tunaweza kuthibitisha vigezo kutoka kwenye mchoro. Kutokana na mchoro, tunaona: • Radiasi ya duara ya kwanza (yenye kituo O_1) ni r_1 = 4 cm. • Radiasi ya duara ya pili (yenye kituo O_2) ni r_2 = 9 cm. • Umbali kati ya vituo vya duara, O_1O_2, ni 9 cm. I) Tafuta urefu wa kodi ya kawaida AB. Step 1: Bainisha vigezo na uanzishe milinganyo. Acha M iwe sehemu ya makutano ya kodi ya kawaida AB na mstari unaounganisha vituo O_1O_2. Kodi ya kawaida AB imegawanywa katika sehemu mbili sawa na M, hivyo AM = MB. Acha AM = h. Acha O_1M = x. Kisha O_2M = O_1O_2 - x = 9 - x. Katika pembetatu ya kulia O_1MA: r_1^2 = O_1M^2 + AM^2 4^2 = x^2 + h^2 16 = x^2 + h^2 (1) Katika pembetatu ya kulia O_2MA: r_2^2 = O_2M^2 + AM^2 9^2 = (9-x)^2 + h^2 81 = (9-x)^2 + h^2 (2) Step 2: Tatua kwa x. Tenga h^2 kutoka mlinganyo (1): h^2 = 16 - x^2. Badilisha h^2 katika mlinganyo (2): 81 = (9-x)^2 + (16 - x^2) 81 = (81 - 18x + x^2) + 16 - x^2 81 = 81 - 18x + 16 0 = -18x + 16 18x = 16 x = (16)/(18) = (8)/(9) cm Step 3: Tatua kwa h na kisha urefu wa kodi AB. Badilisha thamani ya x katika mlinganyo (1): h^2 = 16 - ((8)/(9))^2 h^2 = 16 - (64)/(81) h^2 = (16 × 81 - 64)/(81) h^2 = (1296 - 64)/(81) h^2 = (1232)/(81) Pata mzizi wa mraba wa h^2: h = sqrt((1232)/(81)) = sqrt(1232)sqrt(81) Ili kurahisisha sqrt(1232): 1232 = 16 × 77, hivyo sqrt(1232) = sqrt(16 × 77) = 4sqrt(77). h = 4sqrt(77)9 cm Urefu wa kodi ya kawaida AB ni 2h: AB = 2 × 4sqrt(77)9 = 8sqrt(77)9 cm Urefu wa kodi ya kawaida AB ni 8sqrt(77)9 cm. That's 4 down, 1 left today. What's next?