# The mast

The top of the pole we see at an angle of 45°. If we approach the pole by 10 m, we see the top of the pole at an angle of 60°. What is the height of the pole?

Result

h =  23.66 m

#### Solution:

$\tan(60^\circ ) = h/x \ \\ \tan(45^\circ ) = h/(x+10) \ \\ \ \\ x = h / \tan(60^\circ ) = h \cdot \ t_{ 1 } \ \\ t_{ 1 } = 1/ \tan( (60^\circ \rightarrow rad) = 1/ \tan( (60 \cdot \ \dfrac{ \pi }{ 180 } \ ) = 0.57735026919 ) \ \\ t_{ 2 } = \tan( (45^\circ \rightarrow rad) = \tan( (45 \cdot \ \dfrac{ \pi }{ 180 } \ ) = 1 ) \ \\ \ \\ t_{ 2 } \cdot \ (h \cdot \ t_{ 1 }+10) = h \ \\ t_{ 2 } \cdot \ h \cdot \ t_{ 1 }+10 \cdot \ t_{ 2 } = h \ \\ \ \\ h = \dfrac{ 10 \cdot \ t_{ 2 } }{ 1-t_{ 2 } \cdot \ t_{ 1 } } = \dfrac{ 10 \cdot \ 1 }{ 1-1 \cdot \ 0.5774 } \doteq 23.6603 = 23.66 \ \text { m }$

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