Tree

Between points A and B is 50m. From A we see a tree at an angle 18°. From point B we see the tree in three times bigger angle. How tall is a tree?

Correct result:

h =  13.143 m

Solution:

c=50 m A=18 ∘ B=3⋅ A=3⋅ 18=54 ∘ A0=A∘→ rad=A∘⋅ π180 =18∘⋅ 3.1415926180 =0.31416=π/10 B0=B∘→ rad=B∘⋅ π180 =54∘⋅ 3.1415926180 =0.94248=3π/10 h/c1=tan⁡A h/c2=tan⁡B c=c1+c2 h/tan⁡A+h/tan⁡B=c  h=c/(1/tan⁡(A0)+1/tan⁡(B0))=50/(1/tan⁡(0.3142)+1/tan⁡(0.9425))=13.143 mc=50 \ \text{m} \ \\ A=18 \ ^\circ \ \\ B=3 \cdot \ A=3 \cdot \ 18=54 \ ^\circ \ \\ A_{0}=A ^\circ \rightarrow\ \text{rad}=A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ =18 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ =0.31416=π/10 \ \\ B_{0}=B ^\circ \rightarrow\ \text{rad}=B ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ =54 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ =0.94248=3π/10 \ \\ h/c_{1}=\tan A \ \\ h/c_{2}=\tan B \ \\ c=c_{1}+c_{2} \ \\ h/\tan A + h/ \tan B=c \ \\ \ \\ h=c / (1/\tan(A_{0}) + 1/\tan(B_{0}))=50 / (1/\tan(0.3142) + 1/\tan(0.9425))=13.143 \ \text{m}

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