Answer:
678 feet
Step-by-step explanation:
A big right triangle with a long horizontal leg and a vertical leg length of 141 can be drawn, where the horizontal leg represents the distance between the boat's position at point AA and the bottom of the lighthouse (the vertical leg).
The problem says that at point AA the angle of elevation is 6 degrees meaning that the angle that's opposite of the vertical length has a degree of 6. If this is confusing just think about how the angle "opens upward" like it's being elevated.
The problem then says at point BB the angle of elevation is 12 degrees after some time. If time has passed then the moving boat's distance from the lighthouse got smaller, meaning that a second hypotenuse must be drawn inside the right triangle. The bottom angle will have the angle of 12 degrees and will be point BB.
The length between AA and BB is the desired value, to find that recall SOH-CAH-TOA:
[tex]Sine=\frac{Opposite}{Hypotenuse}[/tex]
[tex]Cosine=\frac{Adjacent}{Hypotenuse}[/tex]
[tex]Tangent=\frac{Opposite}{Adjacent}[/tex].
In this problem since one of the leg's length is known (the lighthouse), and the desire length is along the other leg's length it's best to use the tangent function!
Let the length between AA and BB be d and the distance from BB to the lighthouse be x, then,
[tex]tan(12)=\frac{141}{x}\\\\tan(6)=\frac{141}{d+x}[/tex].
The first equation can be rearranged to find x which can then use it in the second equation to find d!
[tex]x=\frac{141}{tan(12)}[/tex]
Before plugging that fraction into the second equation, that equation can be rearranged to make the calculation look less messy.
[tex]d+x=\frac{141}{tan(6)}[/tex]
[tex]d+\frac{141}{tan(12)} =\frac{141}{tan(6)}[/tex]
[tex]d=678.17=678feet[/tex]
