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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Haya, tutasuluhisha swali hili tena kwa undani zaidi. Swali la 7: Tumia jedwali la logariti kutathmini sqrt((0.0142 × 2.63)/(1.56)). Wacha X = sqrt((0.0142 × 2.63)/(1.56)). Ili kutatua hili kwa kutumia logariti, tunachukua logariti ya pande zote mbili: X = ( ((0.0142 × 2.63)/(1.56))^(1)/(2) ) Tunatumia sheria za logariti: (a^b) = b a, (ab) = a + b, na ((a)/(b)) = a - b. X = (1)/(2) ( (0.0142) + (2.63) - (1.56) ) Step 1: Tafuta logariti za kila namba kutoka jedwali la logariti. • (0.0142): Namba hii ina sifuri mbili kabla ya namba muhimu ya kwanza, hivyo tabia (characteristic) yake ni 2. Mantisa ya 142 ni 0.1523. Hivyo, (0.0142) = 2.1523. • (2.63): Namba hii ina tarakimu moja kabla ya desimali, hivyo tabia yake ni 0. Mantisa ya 263 ni 0.4200. Hivyo, (2.63) = 0.4200. • (1.56): Namba hii ina tarakimu moja kabla ya desimali, hivyo tabia yake ni 0. Mantisa ya 156 ni 0.1931. Hivyo, (1.56) = 0.1931. Step 2: Badilisha thamani hizi kwenye fomula ya X na ufanye hesabu za kujumlisha na kutoa. X = (1)/(2) (2.1523 + 0.4200 - 0.1931) Kwanza, jumlisha logariti za namba kwenye nambari: 2.1523 + 0.4200 = (-2 + 0.1523) + 0.4200 = -2 + (0.1523 + 0.4200) = -2 + 0.5723 = 1.5723 Sasa, toa logariti ya namba kwenye kigawanyo: 1.5723 - 0.1931 = (-1 + 0.5723) - 0.1931 = -1 + (0.5723 - 0.1931) = -1 + 0.3792 = 1.3792 Hivyo, ((0.0142 × 2.63)/(1.56)) = 1.3792. Step 3: Gawanya matokeo kwa 2. X = (1)/(2) (1.3792) Ili kugawanya 1.3792 kwa 2, tunabadilisha tabia 1 ili iwe namba hasi inayoweza kugawanywa na 2 bila salio, huku tukihakikisha mantisa inabaki chanya. 1.3792 = -1 + 0.3792. Tunaweza kuandika -1 kama -2 + 1. Hivyo, 1.3792 = -2 + 1 + 0.3792 = -2 + 1.3792. Sasa gawanya kwa 2: X = (1)/(2) (-2 + 1.3792) = (-2)/(2) + (1.3792)/(2) = -1 + 0.6896 = 1.6896 Step 4: Tafuta antilog ya 1.6896. Mantisa 0.6896 inalingana na takriban 4.893 kutoka jedwali la antilogariti. Tabia 1 inamaanisha kuzidisha kwa 10^-1. X = 4.893 × 10^-1 X = 0.4893 Jibu: 0.4893 That's 2 down. 3 left today — send the next one.