SOLUTION: A woman standing on a hill sees a flagpole that she knows is 60ft tall. The angle of depression to the bottom of the pole is 14, and the angle of elevation to the top of the pole i

Question 413990: A woman standing on a hill sees a flagpole that she knows is 60ft tall. The angle of depression to the bottom of the pole is 14, and the angle of elevation to the top of the pole is 18. Find her distance x from the pole.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Here's a diagram of the problem:
With W being the woman, T being the top of the pole, B being the bottom of the pole and segment WQ being perpendicular to segment TB. WQ is perpendicular to TB because distance from a point, the woman, to a line, the pole, is always measured perpendicularly. You are told that:

The length/height of the pole, TB, is 60 feet.
The angle of elevation, angle TWQ, is 18 degrees.
The angle of depression, angle BWQ, is 14 degrees.

You are asked to find the distance from the woman to the pole, WQ.

Since WQ is perpendicular to TB angles TQW and BQW are right angles and triangles TQW and BQW are right triangles. Since TQW and BQW are right triangles we can form trig ratios with the sides. The trig ratios we will use are those that involve the length we are looking for, WQ, and the sides of the triangles that make up parts of the pole, TQ and BQ. For the angles we know and the sides we want to use, the ratios we will use will be those that involve the opposite and adjacent sides, tan or cot. I will use tan (because I want WQ in the denominator):

and

Multiplying both sides of both equations by WQ we get:
(WQ)*tan(18) = TQ
and
(WQ)*tan(14) = BQ
Now we'll add the two equations together. (You'll see why in a moment.)
(WQ)*tan(18) + (WQ)*tan(14) = TQ + BQ
The right side, TQ + BQ, is the length of the pole which we know is 60. Replacing TQ + BQ with 60 we get:
(WQ)*tan(18) + (WQ)*tan(14) = 60
We can now solve this equation for WQ, the distance from the woman to the pole. Factoring out WQ on the left side we get:
(WQ)*(tan(18) + tan(14)) = 60
Dividing both sides by (tan(18) + tan(14)) we get:

This is an exact expression for the solution to your problem. If you want a decimal approximation of the answer, get out your calculator and find the two tangents and simplify.

If you find the decimal approximation be sure your calculator is in degree mode. If you do not know how to find/set the mode the calculator is in, then

try cos(180). If you get -1 then the calculator is in degree mode and you are ready to find your decimal answer.
try cos(). If you get -1, then you are in radian mode. If you don't know how to switch to degree mode, then convert the degrees to radians:

and use your calculator on this equation.
If cos(180) was not -1 and cos() was not -1, then you are in some other mode. Since I am not familiar with other mode, other than the fact that they exist (one is called gradients, I think), I cannot help you any further other than to say
- Figure out what is equivalent to 180 degrees in another mode
- Find the cos of this number
- If you get -1, then you have found the mode. Then convert the degrees to this other mode. Let's say 200 gradients is equivalent to 180 and cos(200) works out to be -1. Then use the equation:
  
  to find your answer.

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