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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. You're on a roll —Ili kutatua swali hili, tutatumia kanuni za trigonometry. Step 1: Chora mchoro wa hali ilivyo. Hebu H iwe urefu wa mlima. Hebu x iwe umbali wa usawa kutoka sehemu P hadi chini ya mlima. Hebu V iwe sehemu ya kwanza ya uchunguzi. Hebu P iwe sehemu ya pili ya uchunguzi. Umbali kutoka V hadi chini ya mlima ni x + 2000 m. Pembe ya kuinuka kutoka V ni 22^. Pembe ya kuinuka kutoka P ni 40^. Tunatafuta umbali mfupi zaidi kutoka P hadi kilele cha mlima, ambao ni hypotenuse ya pembetatu ya kulia iliyoundwa na P, chini ya mlima, na kilele cha mlima. Hebu tuite umbali huu L. Step 2: Andika milinganyo ya trigonometric. Kutoka sehemu V: (22^) = (H)/(x + 2000) H = (x + 2000)(22^) (Mlinganyo 1) Kutoka sehemu P: (40^) = (H)/(x) H = x(40^) (Mlinganyo 2) Step 3: Tatua kwa x (umbali wa usawa kutoka P). Linganisha Mlinganyo 1 na Mlinganyo 2: (x + 2000)(22^) = x(40^) Panua upande wa kushoto: x(22^) + 2000(22^) = x(40^) Panga upya ili kutenga x: 2000(22^) = x(40^) - x(22^) 2000(22^) = x((40^) - (22^)) Tatua kwa x: x = (2000(22^))/((40^) - (22^)) Tumia thamani za tan: (22^) ≈ 0.4040 na (40^) ≈ 0.8391. x = (2000 × 0.4040)/(0.8391 - 0.4040) x = (808)/(0.4351) x ≈ 1857.04 m Step 4: Tatua kwa H (urefu wa mlima). Tumia Mlinganyo 2 na thamani ya x: H = x(40^) H = 1857.04 × 0.8391 H ≈ 1558.30 m Step 5: Kokotoa umbali mfupi zaidi kutoka P hadi kilele cha mlima (L). Umbali mfupi zaidi ni hypotenuse ya pembetatu ya kulia iliyoundwa na H na x. Tunaweza kutumia (40^) = (H)/(L). L = (H)/((40^)) Tumia thamani ya (40^) ≈ 0.6428. L = (1558.30)/(0.6428) L ≈ 2424.39 m Umbali mfupi zaidi kutoka sehemu P hadi kilele cha mlima ni: 2424.39 m Drop the next question.