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.
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Huu ni mfumo wa kapi wenye mikanda wazi. Kulingana na mchoro na taarifa, angle ya 140^ ni angle isiyogusana na mkanda kwenye kapi kubwa. Data tuliyopewa: Radius ya kapi kubwa, R = 180 cm. Umbali kati ya vituo vya kapi, D = 300 cm. Angle XOY = 140^. Hatua ya 1: Tafuta angle na radius ya kapi ndogo, r. Angle ya kugusana na mkanda kwenye kapi kubwa ni 360^ - 140^ = 220^. Kwa mkanda wazi, angle ya kugusana kwenye kapi kubwa ni _L = 180^ + 2. Hivyo, 220^ = 180^ + 2 2 = 220^ - 180^ 2 = 40^ = 20^ Sasa, tumia formula ya kwa mkanda wazi: = (R-r)/(D) 20^ = (180 - r)/(300) 180 - r = 300 20^ r = 180 - 300 20^ r = 180 - 300 × 0.34202 r = 180 - 102.606 r = 77.394 cm a) Tafuta Urefu XV Urefu XV ni urefu wa sehemu iliyonyooka ya mkanda. XV = D XV = 300 20^ XV = 300 × 0.93969 XV = 281.907 cm Urefu XV ≈ 281.91 cm b) Tafuta Urefu VBW Urefu VBW ni urefu wa arc kwenye kapi ndogo. Angle ya kugusana kwenye kapi ndogo ni _S = 180^ - 2. _S = 180^ - 2(20^) = 180^ - 40^ = 140^ Badilisha angle kuwa radiani: _S = 140^ × ()/(180^) = (7)/(9) radiani Urefu VBW = r _S Urefu VBW = 77.394 × (7)/(9) Urefu VBW ≈ 77.394 × 2.44346 Urefu VBW ≈ 189.195 cm Urefu VBW ≈ 189.20 cm c) Tafuta Urefu XAY Urefu XAY ni urefu wa arc kwenye kapi kubwa. Angle ya kugusana kwenye kapi kubwa ni _L = 220^. Badilisha angle kuwa radiani: _L = 220^ × ()/(180^) = (11)/(9) radiani Urefu XAY = R _L Urefu XAY = 180 × (11)/(9) Urefu XAY = 20 × 11 = 220 cm Urefu XAY ≈ 220 × 3.14159 Urefu XAY ≈ 691.150 cm Urefu XAY ≈ 691.15 cm d) Tafuta Urefu wote wa mkanda wa kusafirisha Urefu wote wa mkanda ni jumla ya sehemu mbili zilizonyooka na urefu wa arc kwenye kila kapi. Urefu wote = 2 × XV + Urefu XAY + Urefu VBW Urefu wote = 2 × 281.907 + 691.150 + 189.195 Urefu wote = 563.814 + 691.150 + 189.195 Urefu wote = 1444.159 cm Urefu wote ≈ 1444.16 cm That's 2 down. 3 left today — send the next one.