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I am having trouble solving this word problem:

A cellular tower that is 150ft is placed on top of a mountain that is 1200 feet above sea level. what is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet above sea level?

Here is my attempt:

-opposite side = 5 miles = 26400ft

-adjacent side = 950 ft

so: tan(?) = (26400ft / 950ft), arctan should give us the angle.

arctan(26400ft/950ft) = 87.94 degrees. this angle is the one with mountain and a slope to the head of user. Thus an angle of depression as it is an angle formed between horizontal line and that slope which is equal to 90 degree -87.94 degrees=2.06 degrees

What am I doing wrong?

Here is a sketch of the problem I made, did I interpret it wrong? enter image description here

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Notice that your answer and the text's answer sum to 90 degrees. The text wants a different, but related, angle from the one you found. –  Matthew Conroy Apr 20 '11 at 23:23
    
So that would be the angle of elevation though, not the angle of depression, wouldn't it? –  Matt Apr 20 '11 at 23:28
    
I think 2.06 degrees is a reasonable description of how depressed the cell phone user's position is from the horizontal line at the top of the cell tower. Your method is absolutely sound: just subtract your result from 90 degrees to get the desired angle. The 87.94 degree angle doesn't measure how elevated anything is, so I wouldn't use the term "angle of elevation" for it, myself. –  Matthew Conroy Apr 20 '11 at 23:36
1  
I quote from www.mathwords.com because I am not a native English speaker: "Angle of Depression The angle below horizontal that an observer must look to see an object that is lower than the observer." mathwords.com/a/angle_depression.htm –  Américo Tavares Apr 20 '11 at 23:41

1 Answer 1

up vote 3 down vote accepted

The angle of elevation refers to the angle above the horizontal. The angle of depression refers to the angle below the horizontal. By the equality of alternate angles, the angle of elevation of the tower top from the cell phone user equals the angle of depression of the cell phone user from the tower top.

The angle labelled in the picture is measured from the vertical, not the horizontal, and so it is neither of these angles.

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