an observer in a lighthouse 300 ft above sea level observes 2 ships directly offshore. The angles of depression to the ships are 3 degrees and 4.5 degrees. How far apart are the two ships?
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change the numbers an proceed. It's exactly the same problem.
To find the distance between the two ships, we need to make use of trigonometry. Here's how we can solve the problem step by step:
Step 1: Draw a diagram:
Draw a diagram representing the situation described. Place the observer at the top of the lighthouse, draw a straight horizontal line to represent the level of the sea, and mark the two ships somewhere on that line.
Step 2: Label the diagram:
Label the observer as O, the lighthouse height as 300 ft, and the distance between the observer and the ships as x. Also, label the two angles of depression as θ1 (for 3 degrees) and θ2 (for 4.5 degrees).
Step 3: Apply trigonometric relationships:
Since we have two right triangles, we can use trigonometric relationships, specifically the tangent function, to find the distances between the observer and the two ships.
For the first ship, we have:
tan(θ1) = opp/adj = (300 ft)/x1
For the second ship, we have:
tan(θ2) = opp/adj = (300 ft)/x2
Step 4: Solve the equations:
Solve each equation for x1 and x2:
x1 = (300 ft) / tan(θ1)
x2 = (300 ft) / tan(θ2)
Step 5: Calculate the distance between the ships:
The distance between the two ships is just the difference between their respective distances from the observer:
Distance = |x1 - x2|
Plug in the values of θ1 and θ2 into the equations obtained in step 4, and calculate x1 and x2. Then, subtract one distance from the other to find the distance between the two ships.
In this case, the values for x1 and x2 are:
x1 = (300 ft) / tan(3 degrees) ≈ 5721.86 ft
x2 = (300 ft) / tan(4.5 degrees) ≈ 4012 ft
Therefore, the distance between the two ships is approximately |5721.86 - 4012| ft ≈ 1709.86 ft.
Hence, the two ships are approximately 1709.86 feet apart.