I am having trouble solving this word problem:
A cellular tower that is $150\text{ ft}$ is placed on top of a mountain that is $1200\text{ ft}$ 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:
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opposite side = $5$ miles = $26400\text{ ft}$
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adjacent side = $950\text{ ft}$
so $$\tan(?) = \frac{26400\text{ ft}} { 950\text{ ft}},$$ and $\arctan$ should give us the angle.
$$\arctan\frac{26400\text{ ft}}{950\text{ ft}} = 87.94^\circ.$$
This angle is the one with mountain and a slope to the head of the user.
Thus an angle of depression as it is an angle formed between the horizontal line and that slope
which is equal to $90^\circ-87.94^\circ = 2.06^\circ$.
What am I doing wrong?
Here is a sketch of the problem I made, did I interpret it wrong?
Best Answer
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.