SOLUTION: Dear Sir/Madam, Please help me with this word problem: From the top of the tower, the angles of depression of the top and bottom of the building 30 meters high are found

Hi, there--

THE PROBLEM:
From the top of the tower, the angles of depression of the top and bottom of the building 30 
meters high are found to be 38 degrees and 56 degrees respectively.  Find the height of the 
tower.

A SOLUTION:
Let h be the height of the tower.
Let x be the distant from the observer to the bottom of the building.

Since both angles are angles of depression, the top of the tower must be higher than the 
building. The angle between the horizontal and the observer's line of sight is the angle of 
depression. 

Refer to the diagram below. The observer stands on top of the tower at point T. The 30m 
building is on the right.

The first angle of depression is formed by the green line extending horizontally from the 
observer and the black line from the observer to the top of the building. The measure of this 
angle is 38 degrees.

The second line of depression is formed by the same green horizontal line and the black line 
extending from the observer to the bottom of the building. The measure of this angle is 56 
degrees. In the diagram, this angle is shown as the sum of two angles measuring 18 and 38
degrees (18+38=56).

Notice the green lines (two parallel lines and one transversal. The (green) angle at the base of 
the building measures 56 degrees because it is an alternate interior angle with the second 
angle of depression. The adjacent angle measures 34 degrees because they are complementary 
angles (their sum is 90 degrees.)

The measure of the interior angle at the top of the building measures 128 degrees, because 
the sum of the interior angles of a triangle is 180 degrees (18+34+128=180.)

Apply the Law of Sines to solve for x.






The distance from the observer to the bottom of the building is 76.5m.

Use the sine of the 56 deg angle at the base of the building to find the h.





The tower is 63.4m tall.




Hope this helps! Please email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com