trigonometry
A woman standing on a hill sees a flagpole that she knows is $35$ ft tall. The angle of depression to the bottom of the pole is $14^{\circ}$, and the angle of elevation to the top of the pole is $18^{\circ}$. Find her distance $x$ from the pole.
So I was able to set it up as follows,
But I don't know how to solve for $x$. What do I do first?
BF is the ground level.
Write EG in terms of angle $50^\circ$.Find $x$. Then the height will be $x\tan 46^\circ + 10$
Define $\,b=$ height of building, $\,f=$height of flag, then basic trigonometry gives
$$\tan 38^\circ=\frac b{250}\implies b=250\tan38^\circ\\{}\\ \tan40^\circ=\frac{f+b}{250}\implies f+b=250\tan40^\circ$$
and thus we get that
$$f=250(\tan40^\circ-\tan38^\circ)=14.453$$
Best Answer
Hint: As you have drawn it, you have two right triangles. You can use the tangent of the angles to evaluate the vertical parts of the flagpole in terms of $x$. They sum to $35$, which will give you an equation for $x$.