Question 11 of 12intermediate🔧 ApplyNumerical4 marks

As observed from the top of a 75 m 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.

Correct Answer

The distance between the two ships is 75(3\sqrt{3} - 1) m or approximately 54.9 m.

Exercise: EXERCISE 9.1 | Q: 13 | (Page Page 10)
For More Understanding

Explanation

This problem involves angles of depression and requires understanding that the angle of depression from the top of the lighthouse equals the angle of elevation from the ship to the top. Since both ships are on the same side with one behind the other, we need to find the distance of each ship from the lighthouse and then subtract to find the distance between them.

Solution Steps

  1. Step 1: Let AB be the lighthouse of height 75 m. Let C be the ship with angle of depression 45° (closer ship), and D be the ship with angle of depression 30° (farther ship).

  2. Step 2: Since the angle of depression equals the angle of elevation from the horizontal, the angle of elevation from ship C to the top A is 45°, and from ship D to A is 30°.

  3. Step 3: In right triangle ABC, apply trigonometric ratio: tan 45° = AB/BC. Since tan 45° = 1, we get BC = 75 m.

  4. Step 4: In right triangle ABD, apply trigonometric ratio: tan 30° = AB/BD. Since tan 30° = 1/3\sqrt{3}, we get BD = 753\sqrt{3} m.

  5. Step 5: Distance between the two ships = BD - BC = 753\sqrt{3} - 75 = 75(3\sqrt{3} - 1) m ≈\approx 54.9 m.

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