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 Dennis, twende kazi! 11. a) Prove that (1+ )/(1+ ) = Step 1: Anza na upande wa kushoto (LHS) na ubadilishe = ( )/( ) na = ( )/( ). LHS = (1+ )/( )1+( )/( ) Step 2: Tafuta denominator ya kawaida kwa nambari ya juu na nambari ya chini. LHS = ( + )/( )( + )/( ) Step 3: Rahisisha kwa kuzidisha nambari ya juu na kinyume cha nambari ya chini. LHS = ( + )/( ) × ( )/( + ) Step 4: Futa maneno yanayofanana ( + ). LHS = ( )/( ) Step 5: Badilisha ( )/( ) na . LHS = Jibu la Mwisho: Kwa kuwa LHS = RHS, uthibitisho umekamilika. (1+ )/(1+ ) = b) Show that ^2 x - ^2 x = 1 - 2^2 x Step 1: Anza na upande wa kushoto (LHS): LHS = ^2 x - ^2 x Step 2: Tumia utambulisho wa Pythagorean ^2 x + ^2 x = 1, ambayo inamaanisha ^2 x = 1 - ^2 x. Step 3: Badilisha ^2 x katika LHS. LHS = (1 - ^2 x) - ^2 x Step 4: Rahisisha usemi. LHS = 1 - 2^2 x Jibu la Mwisho: Kwa kuwa LHS = RHS, uthibitisho umekamilika. ^2 x - ^2 x = 1 - 2^2 x c) Determine the acute angle ^-1 (2.3164) Step 1: Acha = ^-1 (2.3164). Hii inamaanisha = 2.3164. Step 2: Tumia uhusiano = (1)/( ) ili kupata . = (1)/(2.3164) Step 3: Kokotoa thamani ya (1)/(2.3164). ≈ 0.43170436 Step 4: Tumia kazi ya inverse cosine (^-1) kupata . = ^-1(0.43170436) ≈ 64.41^ Jibu la Mwisho: 64.41^ 12. a) Simplify: (3a^-7b^3 × 2ab^-1)/(6a^-1) Step 1: Rahisisha nambari ya juu kwanza kwa kuzidisha coefficients na kuongeza viashiria vya vigezo vinavyofanana. 3a^-7b^3 × 2ab^-1 = (3 × 2) × (a^-7 × a^1) × (b^3 × b^-1) = 6a^(-7+1)b^(3-1) = 6a^-6b^2 Step 2: Gawanya matokeo ya hatua ya 1 na nambari ya chini 6a^-1. 6a^-6b^26a^-1 = ((6)/(6)) × (a^-6a^-1) × b^2 = 1 × a^(-6 - (-1)) × b^2 = a^(-6+1)b^2 = a^-5b^2 Jibu la Mwisho: a^-5b^2 b) Evaluate using scientific calculator correct to 4 decimal places: (3.52)^2 + sqrt(45) Step 1: Kokotoa (3.52)^2. (3.52)^2 = 12.3904 Step 2: Kokotoa sqrt(45). sqrt(45) ≈ 6.708203932 Step 3: Jumlisha matokeo ya hatua ya 1 na 2. 12.3904 + 6.708203932 = 19.098603932 Step 4: Zungusha jibu kwa sehemu 4 za desimali. 19.0986 Jibu la Mwisho: 19.0986 c) A rectangular plot has a length of (x+4) metres and width of (x-2) metres. If the area is 48 m^2, determine the dimensions of the plot. Step 1: Fomula ya eneo la mstatili ni Urefu × Upana. (x+4)(x-2) = 48 Step 2: Panua usemi na upange upya kuwa equation ya quadratic. x^2 - 2x + 4x - 8 = 48 x^2 + 2x - 8 = 48 x^2 + 2x - 8 - 48 = 0 x^2 + 2x - 56 = 0 Step 3: Tatua equation ya quadratic. Tutatumia fomula ya quadratic x = -b ± sqrt(b^2 - 4ac)2a. Hapa a=1, b=2, c=-56. x = -2 ± sqrt(2^2 - 4(1)(-56))2(1) x = -2 ± sqrt(4 + 224)2 x = -2 ± sqrt(228)2 x = (-2 ± 15.09966)/(2) Step 4: Pata maadili mawili ya x. x_1 = (-2 + 15.09966)/(2) = (13.09966)/(2) ≈ 6.5498 x_2 = (-2 - 15.09966)/(2) = (-17.09966)/(2) ≈ -8.5498 Step 5: Kwa kuwa urefu hauwezi kuwa hasi, tunachukua x ≈ 6.55. Step 6: Kokotoa vipimo vya shamba. Urefu = x+4 = 6.55 + 4 = 10.55 m Upana = x-2 = 6.55 - 2 = 4.55 m Jibu la Mwisho: Urefu ≈ 10.55 m, Upana ≈ 4.55 m 13. a) A cone has a radius of 3m and a height of 4m. Calculate its surface area and volume. Step 1: Kokotoa urefu wa slant (l) kwa kutumia Theorem ya Pythagoras. l = sqrt(r^2 + h^2) l = sqrt((3 m))^2 + (4 m)^2 l = sqrt(9 m)^2 + 16 m^2 l = sqrt(25 m)^2 l = 5 m Step 2: Kokotoa eneo la uso (A) la koni. A = r^2 + r l A = (3 m)^2 + (3 m)(5 m) A = 9 m^2 + 15 m^2 A = 24 m^2 Step 3: Kokotoa ujazo (V) wa koni. V = (1)/(3) r^2 h V = (1)/(3) (3 m)^2 (4 m) V = (1)/(3) (9 m^2) (4 m) V = 12 m^3 Jibu la Mwisho: Eneo la uso = 24 m^2, Ujazo = 12 m^3 b) A sphere has a radius of 3m. Finds its Surface area and volume. Step 1: Kokotoa eneo la uso (A) la tufe. A = 4 r^2 A = 4 (3 m)^2 A = 4 (9 m^2) A = 36 m^2 Step 2: Kokotoa ujazo (V) wa tufe. V = (4)/(3) r^3 V = (4)/(3) (3 m)^3 V = (4)/(3) (27 m^3) V = 4 (9 m^3) V = 36 m^3 Jibu la Mwisho: Eneo la uso = 36 m^2, Ujazo = 36 m^3 14. a) Two trucks moving in the same direction on highway, Has the length as follows, Truck A= 12m and Truck B=18m. Given that the speed of truck A is 54 km/h and truck B is 72km/h . How long will truck B take to completely overtake truck A. Step 1: Badilisha kasi kutoka km/h kwenda m/s. (Kumbuka: 1 km/h = (5)/(18) m/s) Kasi ya Lori A (V_A) = 54 km/h = 54 × (5)/(18) m/s = 15 m/s Kasi ya Lori B (V_B) = 72 km/h = 72 × (5)/(18) m/s = 20 m/s Step 2: Kokotoa kasi ya jamaa ya Lori B kwa Lori A. Kasi ya Jamaa (V_rel) = V_B - V_A = 20 m/s - 15 m/s = 5 m/s Step 3: Kokotoa umbali jumla ambao Lori B linahitaji kusafiri ili kulipita kabisa Lori A. Umbali huu ni jumla ya urefu wa malori yote mawili. Umbali (D) = Urefu wa Lori A + Urefu wa Lori B = 12 m + 18 m = 30 m Step 4: Kokotoa muda unaohitajika. Muda (T) = UmbaliKasi ya Jamaa = 30 m5 m/s = 6 s Jibu la Mwisho: 6 sekunde b) A bus travelling at 72 km/h is 200m behind a truck travelling at 54km/h. How will it take the bus to catch up with the truck. Step 1: Badilisha kasi kutoka km/h kwenda m/s. Kasi ya Basi (V_bus) = 72 km/h = 72 × (5)/(18) m/s = 20 m/s Kasi ya Lori (V_truck) = 54 km/h = 54 × (5)/(18) m/s = 15 m/s Step 2: Kokotoa kasi ya jamaa ya basi kwa lori. Kasi ya Jamaa (V_rel) = V_bus - V_truck = 20 m/s - 15 m/s = 5 m/s Step 3: Umbali wa awali kati ya basi na lori ni 200 m. Umbali (D) = 200 m Step 4: Kokotoa muda unaohitajika kwa basi kulifikia lori. Muda (T) = UmbaliKasi ya Jamaa = 200 m5 m/s = 40 s Jibu la Mwisho: 40 sekunde Umefanya vizuri! Tuma swali linalofuata 📸