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.