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 hapa ni ufumbuzi wa swali lako: Sehemu i): Kokotoa urefu wa kamba ya kawaida CD. Step 1: Weka AM = x. Katika pembetatu ya kulia AMC, tunatumia theorem ya Pythagoras: AC^2 = AM^2 + CM^2 5^2 = x^2 + CM^2 CM^2 = 25 - x^2 (Eq. 1) Katika pembetatu ya kulia BMC, BM = AB - AM = 10 - x. Tunatumia theorem ya Pythagoras: BC^2 = BM^2 + CM^2 8^2 = (10-x)^2 + CM^2 CM^2 = 64 - (10-x)^2 (Eq. 2) Step 2: Linganisha Eq. 1 na Eq. 2 ili kupata thamani ya x. 25 - x^2 = 64 - (10-x)^2 25 - x^2 = 64 - (100 - 20x + x^2) 25 - x^2 = 64 - 100 + 20x - x^2 25 = -36 + 20x 25 + 36 = 20x 61 = 20x x = (61)/(20) = 3.05 cm Step 3: Kokotoa CM na kisha CD. Tumia x = 3.05 cm katika Eq. 1: CM^2 = 25 - (3.05)^2 CM^2 = 25 - 9.3025 CM^2 = 15.6975 CM = sqrt(15.6975) ≈ 3.9620 cm Urefu wa kamba ya kawaida CD ni mara mbili ya CM: CD = 2 × CM = 2 × 3.9620 = 7.924 cm Urefu wa kamba ya kawaida CD ni 7.924 cm. Sehemu ii): Kokotoa eneo la eneo lenye kivuli. Eneo lenye kivuli ni jumla ya maeneo ya sehemu mbili za duara (segments) zinazoundwa na kamba CD. Tunatumia ≈ 3.142. Step 4: Kokotoa pembe za katikati (_A na _B). Kwa duara A (radius r_A = 5 cm): Katika AMC: ( CAM) = (AM)/(AC) = (3.05)/(5) = 0.61 CAM = (0.61) ≈ 52.40^ Pembe ya katikati _A = CAD = 2 × CAM = 2 × 52.40^ = 104.80^. Kwa duara B (radius r_B = 8 cm): BM = 10 - x = 10 - 3.05 = 6.95 cm. Katika BMC: ( CBM) = (BM)/(BC) = (6.95)/(8) = 0.86875 CBM = (0.86875) ≈ 29.68^ Pembe ya katikati _B = CBD = 2 × CBM = 2 × 29.68^ = 59.36^. Step 5: Kokotoa eneo la sehemu ya duara A. Eneo la sekta CAD: Eneo la sekta CAD = (_A)/(360^) × r_A^2 = (104.80)/(360) × 3.142 × 5^2 = (104.80)/(360) × 3.142 × 25 ≈ 22.847 cm^2 Eneo la pembetatu CAD: Eneo la CAD = (1)/(2) × CD × AM = (1)/(2) × (7.924) × (3.05) ≈ 12.084 cm^2 Eneo la sehemu ya duara A = Eneo la sekta CAD - Eneo la CAD = 22.847 - 12.084 = 10.763 cm^2 Step 6: Kokotoa eneo la sehemu ya duara B. Eneo la sekta CBD: Eneo la sekta CBD = (_B)/(360^) × r_B^2 = (59.36)/(360) × 3.142 × 8^2 = (59.36)/(360) × 3.142 × 64 ≈ 33.150 cm^2 Eneo la pembetatu CBD: Eneo la CBD = (1)/(2) × CD × BM = (1)/(2) × (7.924) × (6.95) ≈ 27.538 cm^2 Eneo la sehemu ya duara B = Eneo la sekta CBD - Eneo la CBD = 33.150 - 27.538 = 5.612 cm^2 Step 7: Kokotoa jumla ya eneo lenye kivuli. Jumla ya eneo lenye kivuli = Eneo la sehemu ya duara A + Eneo la sehemu ya duara B = 10.763 + 5.612 = 16.375 cm^2 Eneo lenye kivuli ni 16.38 cm^2 (kwa sehemu mbili za desimali). Tuma swali linalofuata 📸