Find the height of the flag.

Question

The height of a flag waving on top of a flag pole is estimated this way: I measure off a distance of 200 ft from the base of the flag pole, and then measure the angle of elevation of the top of the flag, finding it to be 32 degrees. The angle of elevation from the same reference point to the base of the flag is 29.5 degrees. 

Answer 1

BACKGROUND

Firstly, a definition:  the angle of elevation refers to the angle formed by a horizontal line and the line from the object (in this case, the flag) and the observer's eye. 

 

So this problem is giving you enough information to determine the height of the flag itself, by (1) being able to calculate the height of the flag pole corresponding to the top of the flag and (2) being able to calculate the height of the flag pole corresponding to the bottom of the flag.  Once you have these two pieces of information, you would simply subtract the bottom number from the top number.

 

SOLVING THE PROBLEM

 

1)  Draw the figure.  A straight line in the y direction represents the flag pole.  Draw a flag at the top of the pole.  The observer is 200 feet in the x direction away from the pole. 

2) The first angle of elevation is from where the observer is standing to the top of the flag.  We're told this angle is 32 deg.   So now you have a triangle formed by the flag pole, the 200 feet distance to the observer and the line from the observer to the top of the flag.  And it's clearly a right triangle.

3)  You're being asked to find the height of the flag pole to the top of the flag.  Let's call this height y.  So we need to find the opposite side, and we know the adjacent side and the angle.  This means we use tangent function.  Specifically, tan (32) = y / 200. 

4)  Solve the above equation for y.  y = 200* tan(32) = 124.97 ft.

5)  Now we do a similar thing for the bottom of the flag.  Specifically, y = 200 * tan (29.5) = 113.15 ft.

6) The height h of the flag is simply the difference between its top point and bottom point.  So, h = 124.97 - 113.15 = 11.82 ft.