Opposite 78434

We see the tree on the opposite bank of the river at an angle of 15° from a distance of 41m from the river bank. From the bank of the river, we can see at an angle of 31°. How tall is the tree?

Correct answer:

h =  19.83 m

Step-by-step explanation:

$$\begin{gathered} \alpha =15{}^{\circ } \\ a=41\text{m} \\ \beta =31{}^{\circ } \\ \tan \beta =h:r \\ \tan \alpha =h:(r+a) \\ \tan \beta :\tan \alpha =(r+a):r=k \\ k=\tan \beta /\tan \alpha =\tan 31°/\tan 15°=0.600861/0.267949=2.24244 \\ r+a=r\cdot k \\ r+41=r\cdot 2.2424423584781 \\ 1.242442r=41 \\ r=\frac{41}{1.24244236}=32.99951883 \\ r=32.999519 \\ h=r\cdot \tan \beta =r\cdot \tan 31°=32.9995\cdot \tan 31°=32.9995\cdot 0.600861=19.83=19.83\text{m} \end{gathered}$$

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