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)
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Step-by-Step Solution
Step 1: Define variables and initial setup
Let h be the height of the lighthouse, which is given as 100 meters. We will use trigonometry to find the distances from the lighthouse to the ship at two different observation points.
Step 2: Calculate initial distance to the ship
When the angle of depression is 30∘, let the distance of the ship from the lighthouse be x1. Using the tangent function, tan(θ)=adjacentopposite, we find x1=h/tan(30∘).
Step 3: Calculate final distance to the ship
When the angle of depression changes to 45∘, let the new distance of the ship from the lighthouse be x2. Similarly, using the tangent function, we find x2=h/tan(45∘). Since tan(45∘)=1, x2 is equal to h.
Step 4: Determine the distance travelled by the ship
The distance travelled by the ship is the difference between its initial and final distances from the lighthouse, D=x1−x2. Substitute the values of h and 3 to get the numerical answer.