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
Huu hapa ni utatuzi wa swali la 16: a) Tengeneza mlinganyo wa quadratic. Step 1: Fafanua vigezo na uunde mlinganyo kulingana na taarifa uliyopewa. Acha upana wa mstatili uwe W cm. Urefu wa mstatili utakuwa L = (W + 5) cm. Eneo la mstatili A = urefu × upana = (W + 5)W. Tunapewa kuwa eneo A = 40 cm^2. Kwa hiyo, tunaweza kuandika mlinganyo: (W + 5)W = 40 Panua mlinganyo: W^2 + 5W = 40 Panga upya mlinganyo katika fomu ya kawaida ya quadratic ax^2 + bx + c = 0: W^2 + 5W - 40 = 0 b) Tafuta vipimo vya mstatili kwa kuzidisha. Step 2: Jaribu kuzidisha mlinganyo wa quadratic. Mlinganyo ni W^2 + 5W - 40 = 0. Tunahitaji namba mbili ambazo bidhaa yake ni 1 × (-40) = -40 na jumla yake ni 5 (kizigeu cha W). Orodha ya vigawo vya 40 ni (1, 40), (2, 20), (4, 10), (5, 8). Hakuna jozi ya vigawo hivi, hata tukitumia alama zinazofaa, itatoa jumla ya 5. Kwa mfano, 8 × (-5) = -40, lakini 8 + (-5) = 3 ≠ 5. Kwa hivyo, mlinganyo huu wa quadratic hauwezi kuzidishwa katika vigawo vya mstari na vigawo kamili. Step 3: Tatua kwa kutumia fomula ya quadratic kwa kuwa kuzidisha kwa namba kamili haiwezekani. Fomula ya quadratic kwa ax^2 + bx + c = 0 ni x = -b ± sqrt(b^2 - 4ac)2a. Katika mlinganyo wetu, a=1, b=5, na c=-40. W = -5 ± sqrt(5^2 - 4(1)(-40))2(1) W = -5 ± sqrt(25 + 160)2 W = -5 ± sqrt(185)2 Kuna thamani mbili zinazowezekana kwa W: W_1 = -5 + sqrt(185)2 W_2 = -5 - sqrt(185)2 Kwa kuwa upana hauwezi kuwa hasi, tunachukua thamani chanya ya W. W = -5 + sqrt(185)2 cm Takriban W ≈ (-5 + 13.601)/(2) ≈ (8.601)/(2) ≈ 4.30 cm. Step 4: Kokotoa urefu kwa kutumia upana halali. Urefu L = W + 5. L = (-5 + sqrt(185)2) + 5 Ili kuongeza, tunahitaji kupata denominator ya kawaida: L = -5 + sqrt(185)2 + (10)/(2) L = -5 + sqrt(185) + 102 L = 5 + sqrt(185)2 cm Takriban L ≈ (5 + 13.601)/(2) ≈ (18.601)/(2) ≈ 9.30 cm. Kwa hiyo, vipimo vya mstatili ni: Upana = -5 + sqrt(185)2 cm Urefu = 5 + sqrt(185)2 cm That's 2 down. 3 left today — send the next one.