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?
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 $x$.
From left to right(starting at the vertex where the woman is standing), label triangle as ABCD. B is the top of flag, D is on the ground and C is in between B and D.
Let $CD =y$, $BC = 35-y$
$$\tan18^\circ =\frac{35-y}{x}$$ $$ x\tan18^\circ = 35-y$$
$$\tan14^\circ = \frac{y}{x}$$
$$y =x\tan14^\circ$$
Therefore you have a pair of simultaneous equations which you can now solve:
$$ x\tan18^\circ = 35-y$$ $$ x(\tan18^\circ + \tan14^\circ)=35$$
$$x = \frac{35}{\tan18^\circ + \tan14^\circ}$$
Just look at the $18$-degree triangle for a minute. You know one angle and one leg of that triangle. Use those facts and your knowledge of trigonometry to write the length of the other leg.
Now look at the $14$-degree triangle. Again, you know one angle and one leg. Write the length of the other leg.
The "other leg" of the $18$-degree triangle and the "other leg" of the $14$-degree triangle lie end-to-end along the same line (see the figure). That is, they are two pieces of a single line segment. Write a formula for the length of that segment.
The same segment is $35$ feet long according to the figure. So this is equal to your formula. Write the equation.
Solve the equation. Remember there is only one variable in it, so you just need to solve for that variable.
