As observed from the top of a lighthouse, 100 m above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30∘ to 45∘. Determine the distance travelled by the ship during the period of observation. (Use 3=1.732)
(angle of depression, trigonometry, right triangle, distance, lighthouse)
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Step-by-Step Solution
Step 1: Define variables and set up the initial triangle
Let AB be the height of the lighthouse, which is given as 100 m. Let C be the initial position of the ship and D be its final position. The angles of depression are given as 30∘ and 45∘. Due to alternate interior angles, the angle of elevation from the ship to the top of the lighthouse will be equal to the angle of depression. So, ∠ADB=30∘ and ∠ACB=45∘.
Step 2: Calculate distance BC using △ABC
In the right-angled triangle ABC, we can use the tangent function to relate the height of the lighthouse to the distance BC. Since tan(45∘)=1, we find that the distance BC is equal to the height of the lighthouse, 100 m.
Step 3: Calculate distance BD using △ABD
Similarly, in the right-angled triangle ABD, we use the tangent function. We know that tan(30∘)=31. This allows us to calculate the total distance BD from the lighthouse to the ship's initial position.
Step 4: Calculate the distance travelled by the ship
The distance travelled by the ship is the difference between its initial distance from the lighthouse (BD) and its final distance from the lighthouse (BC). We subtract BC from BD to find CD.
Step 5: Substitute the value of 3 and calculate the final distance
Finally, we substitute the given value of 3=1.732 into the expression for CD and perform the arithmetic to get the numerical value of the distance travelled by the ship.