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 utatuzi wa maswali yako: 7. Kiasi P kinatofautiana moja kwa moja na mraba wa Q na kinyume na mzizi wa mraba wa R. Ikiwa P=2 wakati Q=4 na R=16, pata P wakati Q=5 na R=25. Step 1: Andika uhusiano wa utofauti. P inatofautiana moja kwa moja na Q^2 na kinyume na sqrt(R). Hii inaweza kuandikwa kama: P = k (Q^2)/(sqrt(R)) ambapo k ni kigezo cha uwiano. Step 2: Tumia maadili yaliyotolewa kupata thamani ya k. Tunapewa P=2 wakati Q=4 na R=16. 2 = k (4^2)/(sqrt(16)) 2 = k (16)/(4) 2 = k × 4 Gawanya pande zote mbili kwa 4: k = (2)/(4) = (1)/(2) Step 3: Andika fomula kamili ya P. Sasa tuna thamani ya k, fomula inakuwa: P = (1)/(2) (Q^2)/(sqrt(R)) Step 4: Pata P ukitumia maadili mapya. Tunahitaji kupata P wakati Q=5 na R=25. P = (1)/(2) (5^2)/(sqrt(25)) P = (1)/(2) (25)/(5) P = (1)/(2) × 5 P = (5)/(2) Thamani ya P ni (5)/(2) au 2.5. 8. Jedwali hapa chini linaonyesha maadili ya x na y kwa kazi y = ax^2 + bx + c. x: -3, -2, -1, 0, 1, 2, 3 y: 10, 0, -6, -8, -6, 0, 10 a) Tumia jedwali kutatua mlinganyo ax^2 + bx + c = 0. Step 1: Tafuta maadili ya x ambapo y=0 kutoka kwenye jedwali. Mlinganyo ax^2 + bx + c = 0 unamaanisha kutafuta maadili ya x ambapo y ni sifuri. Kutoka kwenye jedwali, tunaona kwamba: Wakati x = -2, y = 0. Wakati x = 2, y = 0. Hivyo, suluhisho za mlinganyo ni x = -2 au x = 2. b) Bainisha maadili ya a, b na c. Step 1: Tumia hatua ambapo x=0 kupata c. Kutoka kwenye jedwali, wakati x=0, y=-8. Badilisha maadili haya kwenye mlinganyo y = ax^2 + bx + c: -8 = a(0)^2 + b(0) + c -8 = 0 + 0 + c c = -8 Step 2: Tumia hatua nyingine mbili kuunda milinganyo ya wakati mmoja kwa a na b. Sasa tunajua c=-8, mlinganyo ni y = ax^2 + bx - 8. Chagua hatua x=1, y=-6: -6 = a(1)^2 + b(1) - 8 -6 = a + b - 8 Ongeza 8 pande zote mbili: a + b = 2 (Mlinganyo 1) Chagua hatua x=2, y=0: 0 = a(2)^2 + b(2) - 8 0 = 4a + 2b - 8 Ongeza 8 pande zote mbili: 4a + 2b = 8 Gawanya mlinganyo mzima kwa 2: 2a + b = 4 (Mlinganyo 2) Step 3: Tatua milinganyo ya wakati mmoja kupata a na b. Toa Mlinganyo 1 kutoka Mlinganyo 2: (2a + b) - (a + b) = 4 - 2 a = 2 Badilisha a=2 kwenye Mlinganyo 1: 2 + b = 2 b = 0 Hivyo, maadili ni a=2, b=0, c=-8. 3 done, 2 left today. You're making progress.