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As observed from the top of a 75 m high...

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.

A

`75(sqrt3-1)`m

B

`85(sqrt3-1)`m

C

`75(sqrt3+1)`m

D

None

Text Solution

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The correct Answer is:
To solve the problem step by step, we will use the properties of right triangles and trigonometric ratios. ### Step 1: Understand the Problem We have a lighthouse of height 75 m. From the top of the lighthouse, the angles of depression to two ships are given as 30° and 45°. We need to find the distance between the two ships. ### Step 2: Draw the Diagram - Let point A be the top of the lighthouse, point B be the base of the lighthouse, point C be the position of the first ship (closer to the lighthouse), and point D be the position of the second ship (farther from the lighthouse). - The height of the lighthouse (AB) = 75 m. - The angle of depression to ship C (angle CAD) = 45°. - The angle of depression to ship D (angle BAD) = 30°. ### Step 3: Use the Angle of Depression Since the angles of depression are equal to the angles of elevation from the ships to the top of the lighthouse: - Angle ACB = 45° (for ship C) - Angle ADB = 30° (for ship D) ### Step 4: Calculate the Distance to Ship C Using triangle ACB: - We have the opposite side (AB) = 75 m and we need to find the adjacent side (CB). - Using the tangent function: \[ \tan(45°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{CB} \] \[ \tan(45°) = 1 \implies 1 = \frac{75}{CB} \implies CB = 75 \text{ m} \] ### Step 5: Calculate the Distance to Ship D Using triangle ADB: - We have the opposite side (AB) = 75 m and we need to find the adjacent side (DB). - Using the tangent function: \[ \tan(30°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{DB} \] \[ \tan(30°) = \frac{1}{\sqrt{3}} \implies \frac{1}{\sqrt{3}} = \frac{75}{DB} \implies DB = 75\sqrt{3} \text{ m} \] ### Step 6: Find the Distance Between the Two Ships The distance between the two ships (CD) can be found as: \[ CD = DB - CB = 75\sqrt{3} - 75 \] Factoring out 75: \[ CD = 75(\sqrt{3} - 1) \text{ m} \] ### Final Answer The distance between the two ships is \( 75(\sqrt{3} - 1) \) meters. ---

To solve the problem step by step, we will use the properties of right triangles and trigonometric ratios. ### Step 1: Understand the Problem We have a lighthouse of height 75 m. From the top of the lighthouse, the angles of depression to two ships are given as 30° and 45°. We need to find the distance between the two ships. ### Step 2: Draw the Diagram - Let point A be the top of the lighthouse, point B be the base of the lighthouse, point C be the position of the first ship (closer to the lighthouse), and point D be the position of the second ship (farther from the lighthouse). - The height of the lighthouse (AB) = 75 m. ...
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