# Depth angles

At the top of the mountain stands a castle, which has a tower 30 meters high. We see the crossroad in the valley from the top of the tower and heel at depth angles of 32° 50 'and 30° 10'. How high is the top of the mountain above the crossroad

Result

y =  272.265 m

#### Solution:

$v=30 \ \text{m} \ \\ \ \\ A=90 - (32 + 50/60)=\dfrac{ 343 }{ 6 } \doteq 57.1667 \ ^\circ \ \\ B=90 - (30 + 10/60)=\dfrac{ 359 }{ 6 } \doteq 59.8333 \ ^\circ \ \\ \ \\ t_{2}=\tan(B)=\tan(59.8333^\circ ) \doteq 1.7205 \ \\ t_{1}=\tan(A)=\tan(57.1667^\circ ) \doteq 1.5497 \ \\ \ \\ \tan A=\dfrac{ x }{ v+y } \ \\ \tan B=x:y \ \\ \ \\ x=y \cdot \ \tan B \ \\ \ \\ y \cdot \ \tan B / \tan A=v+y \ \\ y ( \tan B / \tan A - 1)=v \ \\ \ \\ y=\dfrac{ v }{ t_{2} / t_{1} - 1 }=\dfrac{ 30 }{ 1.7205 / 1.5497 - 1 } \doteq 272.2651 \doteq 272.265 \ \text{m}$

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