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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Huu hapa ni utatuzi wa maswali yako: 4. Fanya s kuwa mada ya fomula a = sqrt((s^2+q)/(p^3)) Step 1: Piga pande zote mbili mraba ili kuondoa mzizi wa mraba. a^2 = (s^2+q)/(p^3) Step 2: Zidisha pande zote mbili kwa p^3. a^2 p^3 = s^2+q Step 3: Toa q kutoka pande zote mbili. a^2 p^3 - q = s^2 Step 4: Piga mzizi wa mraba pande zote mbili ili kumtenga s. s = sqrt(a^2 p^3 - q) Mada ya fomula ni s = sqrt(a^2 p^3 - q). 5. Baada ya t sekunde, chembe inayotembea kwenye mstari ulionyooka ina kasi ya v m/s na kuongeza kasi ya (8 - 2t) m/s^2. Ikiwa kasi ya awali ilikuwa 2 m/s, tambua kasi ya chembe mwanzoni mwa sekunde ya tatu. Step 1: Andika mlinganyo wa kuongeza kasi na uhusiano wake na kasi. Kuongeza kasi, a(t) = (dv)/(dt) = 8 - 2t. Step 2: Unganisha mlinganyo wa kuongeza kasi ili kupata mlinganyo wa kasi. v(t) = (8 - 2t)\,dt v(t) = 8t - (2t^2)/(2) + C v(t) = 8t - t^2 + C Step 3: Tumia kasi ya awali kupata thamani ya C. Kasi ya awali ni v(0) = 2 m/s. 2 = 8(0) - (0)^2 + C C = 2 Hivyo, mlinganyo wa kasi ni v(t) = 8t - t^2 + 2. Step 4: Kokotoa kasi mwanzoni mwa sekunde ya tatu (yaani, wakati t=2 sekunde). v(2) = 8(2) - (2)^2 + 2 v(2) = 16 - 4 + 2 v(2) = 14 m/s Kasi ya chembe mwanzoni mwa sekunde ya tatu ni 14 m/s. 6. Tatua ^2 x - ^2 x = -(1)/(2) kwa 0^ x 360^. Step 1: Tumia utambulisho wa trigonometria (2x) = ^2 x - ^2 x. Mlinganyo uliotolewa ni ^2 x - ^2 x = -(1)/(2). Tunaweza kuandika upya kama -(^2 x - ^2 x) = -(1)/(2). Hii inarahisisha kuwa -(2x) = -(1)/(2). Kuzidisha pande zote mbili kwa -1 kunatoa (2x) = (1)/(2). Step 2: Pata thamani kuu ya 2x. Acha = 2x. Tunahitaji kutatua = (1)/(2). Thamani kuu ni = ^-1((1)/(2)) = 60^. Step 3: Pata thamani zote zinazowezekana za 2x ndani ya masafa yaliyopanuliwa. Kwa kuwa 0^ x 360^, basi 0^ 2x 720^. Cosine ni chanya katika roboduara ya kwanza na ya nne. Thamani zinazowezekana za 2x ni: • 2x = 60^ (Roboduara ya 1) • 2x = 360^ - 60^ = 300^ (Roboduara ya 4) • 2x = 360^ + 60^ = 420^ (Roboduara ya 1 katika mzunguko unaofuata) • 2x = 720^ - 60^ = 660^ (Roboduara ya 4 katika mzunguko unaofuata) Step 4: Tatua kwa x. • 2x = 60^ x = (60^)/(2) = 30^ • 2x = 300^ x = (300^)/(2) = 150^ • 2x = 420^ x = (420^)/(2) = 210^ • 2x = 660^ x = (660^)/(2) = 330^ Thamani za x ni 30^, 150^, 210^, 330^. That's 2 down. 3 left today — send the next one.