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:

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

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