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.
ScanSolve AI Answer
Haya, hapa kuna suluhisho la swali lako: Item 1 a) Kutumia mfumo wa index P(t) = 1200 × (1.15)^t kukokotoa idadi ya samaki mwaka 2025. Step 1: Kokotoa idadi ya miaka t. Mwaka wa kuanzia ni 2020 na mwaka unaotakiwa ni 2025. t = 2025 - 2020 = 5 Step 2: Badilisha t=5 kwenye mfumo. P(5) = 1200 × (1.15)^5 Step 3: Kokotoa thamani. P(5) = 1200 × 2.0113571875 P(5) = 2413.628625 Step 4: Zungusha jibu kwa namba kamili ya samaki. Idadi ya samaki inayotarajiwa mwaka 2025 ni 2414 samaki. b) Kuonyesha kuwa mfumo wa logarithmic ni sawa na mfumo wa index. Mfumo wa index ni P = 1200 × (1.15)^t. Mfumo wa logarithmic ni _10 P = _10 1200 + t _10 1.15. Step 1: Anza na mfumo wa index. P = 1200 × (1.15)^t Step 2: Chukua logarithm ya base 10 pande zote mbili. _10 P = _10 (1200 × (1.15)^t) Step 3: Tumia sheria ya logarithm (AB) = A + B. _10 P = _10 1200 + _10 (1.15)^t Step 4: Tumia sheria ya logarithm (A^B) = B A. _10 P = _10 1200 + t _10 1.15 Hii inaonyesha kuwa mfumo wa logarithmic ni sawa na mfumo wa index. c) Kuamua idadi ya miaka (t) itakayochukua kwa idadi ya samaki kufikia 5,000. Step 1: Weka P(t) = 5000 kwenye mfumo wa index. 5000 = 1200 × (1.15)^t Step 2: Gawanya pande zote mbili kwa 1200. (5000)/(1200) = (1.15)^t (25)/(6) = (1.15)^t 4.1666... = (1.15)^t Step 3: Chukua logarithm ya base 10 pande zote mbili. _10 ((25)/(6)) = _10 ((1.15)^t) Step 4: Tumia sheria ya logarithm (A^B) = B A. _10 ((25)/(6)) = t _10 (1.15) Step 5: Tatua kwa t. t = _10 ((25)/(6))_10 (1.15) Step 6: Kokotoa thamani za logarithm na t. t = (0.619789)/(0.0606978) t ≈ 10.21 Idadi ya miaka itakayochukua ni takriban 10.2 miaka. Tuma swali linalofuata 📸