SOLUTION: from the top of a 25m lighthouse, an operator sees a capsized boat and determines an angle of depression of 7 degrees to the boat. A patrol boat is also spotted at an angle of depr

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Question 980116: from the top of a 25m lighthouse, an operator sees a capsized boat and determines an angle of depression of 7 degrees to the boat. A patrol boat is also spotted at an angle of depression of 5 degrees.(a)If the two boats are on the same side of the lighthouse, how far apart are the two boats? (b)If the boats are on opposite sides of the lighthouse, how far apart are the two boats?
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
The two separate distanaces from top of the lighthouse to the boats form both transversals of two parallel lines as well as hypotenuses of two right triangles. The angles of depression are also measures of angles of elevation AT each of the boats.

I will answer the second question.

You have lighthouse height, angle opposite, a right angle at base of lighthouse, and you want to find the unknown legs of the two triangles. The two unknown legs have endpoints extending from one boat to the other boat.

You can and SHOULD make the drawing.

One triangle:
b for unknown leg,
tan%285%29=25%2Fb
b%2F25=1%2Ftan%285%29
b=25%2Ftan%285%29

Other triangle:
x for unknown leg,
tan%287%29=25%2Fx
x%2F25=1%2Ftan%287%29
x=25%2Ftan%28x%29

Distance between the boats,
b%2Bx
highlight%28highlight%2825%2Ftan%285%29%2B25%2Ftan%287%29%29%29

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

from the top of a 25m lighthouse, an operator sees a capsized boat and determines an angle of depression of 7 degrees to the boat. A patrol boat is also spotted at an angle of depression of 5 degrees.(a)If the two boats are on the same side of the lighthouse, how far apart are the two boats? (b)If the boats are on opposite sides of the lighthouse, how far apart are the two boats?
Distance capsized boat is from lighthouse: highlight_green%28203.61%29 m 

Distance patrol boat is from lighthouse: highlight_green%28285.751%29 m

Distance boats are from each other: 281.751 - 203.61, or highlight_green%2882.141%29 m