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A woman looking out from the window of a height of 30m, observed that the angle of depression of the top of a flag pole was 44 degrees. If the foot of the pole is 25m from the foot of the building and on the same horizontal ground, find, correct to the nearest whole number, the angle of depression of the foot of the pole from the woman and the height of the flag pole

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Check the picture below.

so the height of the pole is "p", and the angle of depression to the foot of the pole is really "44 + a".

so let's use the 44° angle first

[tex]\tan(44^o )=\cfrac{\stackrel{opposite}{30-p}}{\underset{adjacent}{25}}\implies 25\tan(44^o)=30-p\implies 25\tan(44^o)+p=30 \\\\\\ p=30-25\tan(44^o)\implies \boxed{p\approx 9}[/tex]

now, let's use the red triangle, namely the angle "44 + a"

[tex]\tan(44+a)=\cfrac{\stackrel{opposite}{30}}{\underset{adjacent}{25}}\implies \tan(44+a)=\cfrac{6}{5} \\\\\\ 44+a=\tan^{-1}\left( \cfrac{6}{5} \right) \implies 44+a\approx 50^o[/tex]

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