# Thread: angle depression

1. ## angle depression

A light house keeper 120 feet above the water sees a boat sailing in a straight line directly toward her. As she watches, the angle of depression to the boat changes from 28 degrees to 43 degrees. How far has the boat traveled during this time?

2. Hello, victorfk06!

A lighthouse keeper 120 feet above the water sees a boat
sailing in a straight line directly toward her.
As she watches, the angle of depression to the boat changes from 28° to 43°.
How far has the boat traveled during this time?
Code:
    A * - - - - - - - - - - -
|   *  28°
| 62°  *
|         *
120 |             *
|               *
|                  *
|                     *
|                        *
* - - - -*- - - - - - - - - *
B   y    C        x         D
In right triangle $\displaystyle ABD\!:\;\;\tan62^o \:=\:\frac{x+y}{120}\quad\Rightarrow\quad x \;=\;120\tan62^o - y$ .[1]

Code:
    A * - - - - - -
|* 43°
| *
|  *
120 |47°*
|    *
|     *
|      *
|       *
* - - - -*
B   y    C
In right triangle $\displaystyle ABC\!:\;\;\tan47^o \:=\:\frac{y}{120} \quad\Rightarrow\quad y \:=\:120\tan47^o$ .[2]

Substitute [2] into [1]: .$\displaystyle x \;=\;120\tan62^o - 120\tan47^o \;=\;97.0029306$

. . Therefore, the boat traveled about 97 feet.

3. There are various approaches, but the law of tangents works OK.

$\displaystyle 120(tan(62)-tan(47))$