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: Kurahisisha usemi Ili kurahisisha usemi sqrt(5)+3sqrt(5)-2, tunazidisha nambari na denominata kwa kiambishi cha denominata, ambacho ni sqrt(5)+2. sqrt(5)+3sqrt(5)-2 × sqrt(5)+2sqrt(5)+2 Step 2: Kupanua nambari na denominata Panua nambari: (sqrt(5)+3)(sqrt(5)+2) = sqrt(5)·sqrt(5) + sqrt(5)·2 + 3·sqrt(5) + 3·2 = 5 + 2sqrt(5) + 3sqrt(5) + 6 = 11 + 5sqrt(5) Panua denominata (kumbuka (a-b)(a+b) = a^2-b^2): (sqrt(5)-2)(sqrt(5)+2) = (sqrt(5))^2 - (2)^2 = 5 - 4 = 1 Step 3: Kuweka pamoja na kurahisisha Sasa weka nambari na denominata pamoja: 11 + 5sqrt(5)1 = 11 + 5sqrt(5) Hii iko katika mfumo a + bsqrt(c) ambapo a=11, b=5, na c=5. 11 + 5sqrt(5) 2. Uwiano wa unga X:Y Step 1: Kuunda mlinganyo Acha x iwe kiasi cha unga X na y iwe kiasi cha unga Y. Gharama ya unga X ni Ksh 60 kwa kg, na unga Y ni Ksh 72 kwa kg. Gharama ya mchanganyiko ni Ksh 70 kwa kg. Gharama ya jumla ya mchanganyiko ni 60x + 72y. Kiasi cha jumla cha mchanganyiko ni x + y. Gharama kwa kg ya mchanganyiko ni (60x + 72y)/(x + y) = 70. Step 2: Kutatua mlinganyo kwa uwiano x:y (60x + 72y)/(x + y) = 70 Zidisha pande zote mbili kwa (x+y): 60x + 72y = 70(x + y) 60x + 72y = 70x + 70y Panga upya vigezo: 72y - 70y = 70x - 60x 2y = 10x Ili kupata uwiano x:y, gawanya pande zote mbili kwa y na kwa 10: (2y)/(10y) = (10x)/(10y) (2)/(10) = (x)/(y) (1)/(5) = (x)/(y) Hivyo, uwiano x:y ni 1:5. 1:5 3. Kuamua P wakati L = 0.2 Step 1: Kuunda mlinganyo wa utofauti Kiasi P kinatofautiana kinyume na mraba wa kiasi kingine L. Hii inaweza kuandikwa kama: P = (k)/(L^2) ambapo k ni kigezo cha utofauti. Step 2: Kupata thamani ya kigezo k Tunapewa P = 0.625 wakati L = 4. Badilisha hizi kwenye mlinganyo: 0.625 = (k)/(4^2) 0.625 = (k)/(16) Ili kupata k, zidisha pande zote mbili kwa 16: k = 0.625 × 16 k = 10 Step 3: Kuamua P wakati L = 0.2 Sasa tunajua k = 10. Tunataka kupata P wakati L = 0.2. Badilisha hizi kwenye mlinganyo wa utofauti: P = (10)/((0.2)^2) P = (10)/(0.04) P = (10)/(4)100 P = 10 × (100)/(4) P = (1000)/(4) P = 250 250 Tuma swali linalofuata 📸