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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Step 1: Andika uhusiano wa utofauti. B inatofautiana kwa sehemu kama mraba wa M (M^2) na kwa sehemu kama kinyume cha N ((1)/(N)). Hii inamaanisha tunaweza kuandika uhusiano kama: B = k_1 M^2 + (k_2)/(N) ambapo k_1 na k_2 ni vigezo vya uwiano. Step 2: Tumia hali ya kwanza kuunda mlinganyo. Tunapewa M=2, N=(1)/(2), na B=76. Badilisha hizi kwenye fomula: 76 = k_1 (2)^2 + (k_2)/(1)2 76 = 4k_1 + 2k_2 (Mlinganyo 1) Step 3: Tumia hali ya pili kuunda mlinganyo. Tunapewa sqrt(M)=3, N=2, na B=46. Kutoka sqrt(M)=3, tunapata M = 3^2 = 9. Badilisha M=9, N=2, na B=46 kwenye fomula: 46 = k_1 (9)^2 + (k_2)/(2) 46 = 81k_1 + (1)/(2)k_2 (Mlinganyo 2) Step 4: Tatua mfumo wa milinganyo ili kupata k_1 na k_2. Kutoka Mlinganyo 1, gawanya kwa 2: 38 = 2k_1 + k_2 k_2 = 38 - 2k_1 Badilisha k_2 kwenye Mlinganyo 2: 46 = 81k_1 + (1)/(2)(38 - 2k_1) 46 = 81k_1 + 19 - k_1 46 - 19 = 80k_1 27 = 80k_1 k_1 = (27)/(80) Sasa tafuta k_2: k_2 = 38 - 2((27)/(80)) k_2 = 38 - (27)/(40) k_2 = (38 × 40 - 27)/(40) k_2 = (1520 - 27)/(40) k_2 = (1493)/(40) Step 5: Andika usemi wa B kwa kutumia k_1 na k_2. Badilisha thamani za k_1 na k_2 kwenye fomula ya awali: B = (27)/(80) M^2 + (1493)/(40N) Usemi wa B kwa kutumia M na N ni: B = (27)/(80) M^2 + (1493)/(40N) That's 2 down. 3 left today — send the next one.