Question

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships as 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is

• A. 300 m
• B. 200 m
• C. 173 m
• D. 273 m

Hint:

For problems involving angles of elevation, use the tangent function to relate the height and distance from the object.

Answer 1

The Correct Option is D

Step 1: Let the height of the lighthouse be \( h = 100 \) m. Let the distance between the lighthouse and the ships be \( x_1 \) and \( x_2 \), corresponding to the angles of elevation of 30° and 45°, respectively. Step 2: For the ship observing the lighthouse at 30°, we can use the tangent of the angle of elevation: \[ \tan(30^\circ) = \frac{h}{x_1} \] Substituting the values: \[ \frac{1}{\sqrt{3}} = \frac{100}{x_1} \] \[ x_1 = 100\sqrt{3} \approx 173.2 \, \text{m} \] Step 3: For the ship observing the lighthouse at 45°, we use: \[ \tan(45^\circ) = \frac{h}{x_2} \] \[ 1 = \frac{100}{x_2} \] \[ x_2 = 100 \, \text{m} \] Step 4: The total distance between the two ships is the sum of \( x_1 \) and \( x_2 \): \[ \text{Distance} = x_1 + x_2 = 173.2 + 100 = 273.2 \, \text{m} \] Thus, the distance between the two ships is approximately 273 m.