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 General, twende kazi! Hili ni swali la jiometri ya 3D. Tutatumia mfumo wa kuratibu (coordinates) kutatua. Kwanza, hebu tueleze kuratibu za pembe za mchemraba (cuboid) PQRSTUVW. Kutokana na mchoro na vipimo vilivyotolewa (PQ = 9 cm, PS = 5 cm, PT = 12 cm), tutachukulia P kama asili (origin) (0,0,0). P = (0,0,0) Q = (9,0,0) (urefu wa PQ) S = (0,5,0) (upana wa PS) T = (0,0,12) (urefu wa PT) R = (9,5,0) (Q + S - P) U = (9,0,12) (Q + T - P) V = (0,5,12) (S + T - P) W = (9,5,12) (R + T - P) Kumbuka: Maelezo ya swali "length PR = 9m" yanapingana na mchoro unaoonyesha PQ = 9cm kama urefu wa upande. Tutatumia vipimo vya mchoro (PQ=9, PS=5, PT=12) kwa sababu ndivyo vinavyoendana na umbo la mchemraba. i) Kokotoa pembe kati ya mstari PU na msingi PQRS. Mstari ni PU. Msingi ni ndege PQRS (ndege z=0). Makadirio ya PU kwenye ndege PQRS ni mstari PQ. Hivyo, pembe kati ya mstari PU na ndege PQRS ni pembe UPQ. Tuna pembetatu mraba UPQ. U = (9,0,12) P = (0,0,0) Q = (9,0,0) Urefu wa PQ = 9 cm. Urefu wa UQ = PT = 12 cm. Katika UPQ: ( UPQ) = OppositeAdjacent = (UQ)/(PQ) = (12)/(9) = (4)/(3) UPQ = ((4)/(3)) UPQ ≈ 53.1301^ Ikiwa imezungushwa hadi sehemu moja ya desimali: 53.1^ ii) Kokotoa pembe kati ya mstari SU na ndege WTUV. Mstari ni SU. Ndege ni WTUV (ndege z=12, uso wa juu). Makadirio ya S kwenye ndege WTUV ni V (kwa sababu SV iko wima kwa ndege WTUV). Makadirio ya U kwenye ndege WTUV ni U yenyewe (kwa sababu U iko kwenye ndege). Hivyo, makadirio ya mstari SU kwenye ndege WTUV ni mstari VU. Pembe kati ya mstari SU na ndege WTUV ni pembe SUV. Tuna pembetatu mraba SVU. S = (0,5,0) U = (9,0,12) V = (0,5,12) Urefu wa SV = PT = 12 cm. Urefu wa VU = PQ = 9 cm. Katika SVU: ( SUV) = OppositeAdjacent = (SV)/(VU) = (12)/(9) = (4)/(3) SUV = ((4)/(3)) SUV ≈ 53.1301^ Ikiwa imezungushwa hadi sehemu moja ya desimali: 53.1^ iii) Kokotoa pembe kati ya mstari AV na ndege RQUV, ambapo A ni sehemu ya kati ya PQ. Kwanza, pata kuratibu za A. P = (0,0,0), Q = (9,0,0). A ni sehemu ya kati ya PQ: A = ((0+9)/(2), (0+0)/(2), (0+0)/(2)) = (4.5, 0, 0) Mstari ni AV. V = (0,5,12). Vekta ya mwelekeo wa mstari AV ni: l = AV = V - A = (0 - 4.5, 5 - 0, 12 - 0) = (-4.5, 5, 12) Ndege ni RQUV. R = (9,5,0) Q = (9,0,0) U = (9,0,12) V = (0,5,12) Kumbuka: Pointi Q, R, U, V haziko kwenye ndege moja. Hii inafanya swali kuwa na utata. Tutachukulia "ndege RQUV" inamaanisha ndege inayopita kwenye pointi Q, R, na U, ambayo ni ndege ya uso wa kulia wa mchemraba (plane x=9). Ikiwa ndege ni x=9: Vekta ya kawaida (normal vector) ya ndege x=9 ni n = (1, 0, 0). Pembe kati ya mstari na ndege inatolewa na formula: = |l · n|||l|| · ||n|| Kokotoa dot product l · n: l · n = (-4.5)(1) + (5)(0) + (12)(0) = -4.5 Kokotoa ukubwa wa vekta l: ||l|| = sqrt((-4.5)^2 + 5^2 + 12^2) = sqrt(20.25 + 25 + 144) = sqrt(189.25) Kokotoa ukubwa wa vekta n: ||n|| = sqrt(1^2 + 0^2 + 0^2) = sqrt(1) = 1 Sasa tumia formula: = (|-4.5|)/(sqrt(189.25) × 1) = (4.5)/(sqrt(189.25)) = (4.5)/(13.756816...) ≈ 0.32712 = (0.32712) ≈ 19.09^ Ikiwa imezungushwa hadi sehemu moja ya desimali: 19.1^ 3 done, 2 left today. You're making progress.