# 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?

Correct result:

h =  23.66 m

#### Solution:

$\tan(60^\circ )=h/x \ \\ \tan(45^\circ )=h/(x+10) \ \\ \ \\ x=h / \tan(60^\circ )=h \cdot \ t_{1} \ \\ t_{1}=1/ \tan 60 ^\circ =\tan π/3=0.57735 \ \\ t_{2}=\tan 45 ^\circ =\tan π/4=1 \ \\ \ \\ t_{2} \cdot \ (h \cdot \ t_{1}+10)=h \ \\ t_{2} \cdot \ h \cdot \ t_{1}+10 \cdot \ t_{2}=h \ \\ \ \\ h=\dfrac{ 10 \cdot \ t_{2} }{ 1-t_{2} \cdot \ t_{1} }=\dfrac{ 10 \cdot \ 1 }{ 1-1 \cdot \ 0.5774 }=23.66 \ \text{m}$

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