From the top of a 90 m lighthouse, an operator sees a capsized boat and determines an angle of depression of
12.5° to the boat. A patrol boat is also spotted at an angle of depression of 9°.
If the two boats are on the opposite side of the lighthouse, how far apart are the two boats?
Let's assume that the distance between the two boats is x meters.
From the top of the lighthouse, the operator sees the capsized boat and determines an angle of depression of 12.5°. This means that the angle between the horizontal line and the line connecting the operator and the capsized boat is 12.5°.
Using trigonometry, we can find the height of the capsized boat as follows:
tan(12.5°) = height of capsized boat / 90 m
height of capsized boat = tan(12.5°) * 90 m
Similarly, the angle of depression of the patrol boat is 9°. This means that the angle between the horizontal line and the line connecting the operator and the patrol boat is 9°.
Using trigonometry, we can find the height of the patrol boat as follows:
tan(9°) = height of patrol boat / 90 m
height of patrol boat = tan(9°) * 90 m
Since the two boats are on the opposite sides of the lighthouse, the total distance between them is equal to the sum of their individual heights:
Distance between the two boats = height of capsized boat + height of patrol boat
Distance between the two boats = tan(12.5°) * 90 m + tan(9°) * 90 m
Distance between the two boats = (tan(12.5°) + tan(9°)) * 90 m
Calculating this expression gives the distance between the two boats.
To find the distance between the two boats, we can use trigonometry.
Let's call the distance between the lighthouse and the capsized boat "x1" and the distance between the lighthouse and the patrol boat "x2".
We can use the tangent function to find the distances:
tangent(angle of depression) = opposite / adjacent
For the capsized boat:
tangent(12.5°) = 90 / x1
For the patrol boat:
tangent(9°) = 90 / x2
To find the distance between the two boats, we can subtract the two distances:
Distance between the two boats = x1 - x2
Now, let's calculate the distances:
Using the tangent function for the capsized boat:
tangent(12.5°) = 90 / x1
Rearranging the equation to solve for x1:
x1 = 90 / tangent(12.5°)
Calculating x1:
x1 = 90 / 0.2227
x1 ≈ 404.34 meters
Using the tangent function for the patrol boat:
tangent(9°) = 90 / x2
Rearranging the equation to solve for x2:
x2 = 90 / tangent(9°)
Calculating x2:
x2 = 90 / 0.1584
x2 ≈ 567.42 meters
Now we can find the distance between the two boats:
Distance between the two boats = x1 - x2
Distance between the two boats ≈ 404.34 - 567.42
Distance between the two boats ≈ -163.08 meters
Since the distance is negative, it means that the two boats are in opposite directions from the lighthouse. Therefore, the distance between the two boats is approximately 163.08 meters.