A man is standing on the deck of a ship which is 25 m above water level. he observes the angle of elevation of the top of a lighthouse as 60 degrees and the angle of depression of the base of the lighthouse as 45 degrees. calculate the height of the lighthouse. (lass 10, CBSE board, India)
Question
A man is standing on the deck of a ship which is 25 m above water level. he observes the angle of elevation of the top of a lighthouse as 60 degrees and the angle of depression of the base of the lighthouse as 45 degrees. calculate the height of the lighthouse. (lass 10, CBSE board, India)
Solution
To solve this problem, we will use trigonometry. Specifically, we will use the tangent of the angles given.
Step 1: Let's denote the height of the lighthouse as H (in meters), the distance between the ship and the lighthouse as D (in meters), and the height of the ship as 25 m.
Step 2: From the problem, we know that the angle of elevation to the top of the lighthouse is 60 degrees. This means that tan(60) = (H+25)/D.
Step 3: We also know that the angle of depression to the base of the lighthouse is 45 degrees. This means that tan(45) = 25/D.
Step 4: We can solve the equation from step 3 for D, which gives us D = 25/tan(45) = 25 m.
Step 5: Substituting D = 25 m into the equation from step 2 gives us tan(60) = (H+25)/25.
Step 6: Solving this equation for H gives us H = 25tan(60) - 25 = 25sqrt(3) - 25 = 43.3 m (approx).
So, the height of the lighthouse is approximately 43.3 m.
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