Observer

The observer sees a straight fence 100 m long in 30° view angle. From one end of the fence is 119 m.

How far is it from the other end of the fence?

Correct answer:

x =  139.9 m

Step-by-step explanation:

a=119 m l=70 m α=30 ∘  sin⁡ β sin⁡ α = al sin⁡ β = al⋅ sin⁡ α = s  s=al⋅ sin⁡α=al⋅ sin⁡30° =11970⋅ sin⁡30° =11970⋅ 0.5=0.85 β=180°π⋅arcsin⁡s=180°π⋅arcsin⁡0.85≐58.2117  γ=180−α−β=180−30−58.2117≐91.7883 ∘  sin⁡ γ sin⁡ β=xa  x=a⋅ sin⁡γsin⁡β=a⋅ sin⁡91.788330617052° sin⁡58.211669382948° =119⋅ sin⁡91.788330617052° sin⁡58.211669382948° =119⋅ 0.9995130.85=139.9=139.9 ma = 119 \ \text{m} \ \\ l = 70 \ \text{m} \ \\ α = 30 \ ^\circ \ \\ \ \\ \dfrac{ \sin\ β }{ \ \sin\ α } \ = \ \dfrac{ a }{ l } \ \\ \sin\ β\ = \ \dfrac{ a }{ l } \cdot \ \sin\ α\ = \ s \ \\ \ \\ s = \dfrac{ a }{ l } \cdot \ \sin α = \dfrac{ a }{ l } \cdot \ \sin 30\degree \ = \dfrac{ 119 }{ 70 } \cdot \ \sin 30\degree \ = \dfrac{ 119 }{ 70 } \cdot \ 0.5 = 0.85 \ \\ β = \dfrac{ 180\degree }{ \pi } \cdot \arcsin s = \dfrac{ 180\degree }{ \pi } \cdot \arcsin 0.85 \doteq 58.2117 \ \\ \ \\ γ = 180-α-β = 180-30-58.2117 \doteq 91.7883 \ ^\circ \ \\ \ \\ \dfrac{ \sin\ γ }{ \ \sin\ β } = \dfrac{ x }{ a } \ \\ \ \\ x = a \cdot \ \dfrac{ \sin γ }{ \sin β } = a \cdot \ \dfrac{ \sin 91.788330617052\degree \ }{ \sin 58.211669382948\degree \ } = 119 \cdot \ \dfrac{ \sin 91.788330617052\degree \ }{ \sin 58.211669382948\degree \ } = 119 \cdot \ \dfrac{ 0.999513 }{ 0.85 } = 139.9 = 139.9 \ \text{m}

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You need to know the following knowledge to solve this word math problem:

algebraarithmeticplanimetricsbasic operations and conceptsnumbersgoniometry and trigonometryUnits of physical quantitiesGrade of the word problem

 
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