Depth angles

At the top of the mountain stands a castle with a tower 30 meters high. We see the crossroad at a depth angle of 32°50' and the heel at 30°10' from the top of the tower. How high is the top of the mountain above the crossroad?

Correct answer:

y =  272.2651 m

Step-by-step explanation:

$$\begin{gathered} v=30\text{m} \\ A=90-(32+50/60)=\frac{343}{6}\doteq 57.1667{}^{\circ } \\ B=90-(30+10/60)=\frac{359}{6}\doteq 59.8333{}^{\circ } \\ {t}_2=\tan (B)=\tan (59.8333°)\doteq 1.7205 \\ {t}_1=\tan (A)=\tan (57.1667°)\doteq 1.5497 \\ \tan A=\frac{x}{v+y} \\ \tan B=x:y \\ x=y\cdot \tan B \\ y\cdot \tan B/\tan A=v+y \\ y(\tan B/\tan A-1)=v \\ y=\frac{v}{t_2/t_1-1}=\frac{30}{1.7205/1.5497-1}=272.2651\text{m} \end{gathered}$$

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