As observed from the top of a 100m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
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Step-by-Step Solution
Step 1: Define variables and set up the first triangle
Let AB be the height of the lighthouse, which is 100 m. Let C be the position of the first ship. The angle of depression to the first ship is 45∘, so the angle of elevation from the ship to the top of the lighthouse, ∠ACB, is also 45∘. We use the tangent function in the right-angled triangle ABC to relate the height of the lighthouse to the distance BC.
Step 2: Calculate distance to the first ship
Since tan45∘=1, we can substitute this value into our equation. This allows us to calculate the distance BC, which is the distance from the base of the lighthouse to the first ship.
Step 3: Set up the second triangle
Now, let D be the position of the second ship. The angle of depression to the second ship is 30∘, so the angle of elevation from the ship to the top of the lighthouse, ∠ADB, is also 30∘. We use the tangent function in the right-angled triangle ABD to relate the height of the lighthouse to the distance BD.
Step 4: Calculate distance to the second ship
Since tan30∘=31, we substitute this value into the equation. This allows us to calculate the distance BD, which is the distance from the base of the lighthouse to the second ship.
Step 5: Find the distance between the two ships
The distance between the two ships, CD, is the difference between the distance to the second ship (BD) and the distance to the first ship (BC). We substitute the calculated values to find the final distance.