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Anil looked up at the top of a lighthouse from his boat and found the angle of elevation to be 30 degrees. After sailing in a straight line 50 m towards the lighthouse, he found that the angle of elevation changed to 45 degrees. Find the height of the lighthouse.
- a)25
- b)25√3
- c)25(√3-1)
- d)25(√3+1)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Anil looked up at the top of a lighthouse from his boat and found the ...
If we look at the above image, A is the previous position of the boat. Angle of elevation from this point to the top of the light house is 30 degrees.
After sailing for 50 m, Anil reaches point D from where angle of elevation is 45 degrees. C is the top of the light house.
Let BD = x
Now, we know tan 30 degrees = 1/ √3 = BC/AB
Tan 45 degrees = 1
=> BC = BD = x
Thus, 1/ √3 = BC/AB = BC / (AD+DB) = x / (50 + x)
Thus x (√3 -1) = 50 or x= 25(√3 +1) m
The answer is Option D.
Most Upvoted Answer
Anil looked up at the top of a lighthouse from his boat and found the ...
Let's assume that the height of the lighthouse is "h" meters.
When Anil is at point A (on his boat), the angle of elevation to the top of the lighthouse is 30 degrees.
When Anil sails 50 meters towards the lighthouse and reaches point B, the angle of elevation to the top of the lighthouse is 45 degrees.
We can form a right-angled triangle ABC, where AB is the distance Anil sailed towards the lighthouse (50 meters), BC is the height of the lighthouse (h meters), and angle BAC is 30 degrees.
Using the tangent function, we can write:
tan(30) = BC / AB
tan(30) = h / 50
Solving for h, we get:
h = 50 * tan(30)
h = 50 * (1/√3)
h = 50/√3
h = (50/√3) * (√3/√3) [Multiplying numerator and denominator by √3]
h = (50√3) / 3
So, the height of the lighthouse is approximately 28.87 meters.
Therefore, the correct option is:
b) 25
When Anil is at point A (on his boat), the angle of elevation to the top of the lighthouse is 30 degrees.
When Anil sails 50 meters towards the lighthouse and reaches point B, the angle of elevation to the top of the lighthouse is 45 degrees.
We can form a right-angled triangle ABC, where AB is the distance Anil sailed towards the lighthouse (50 meters), BC is the height of the lighthouse (h meters), and angle BAC is 30 degrees.
Using the tangent function, we can write:
tan(30) = BC / AB
tan(30) = h / 50
Solving for h, we get:
h = 50 * tan(30)
h = 50 * (1/√3)
h = 50/√3
h = (50/√3) * (√3/√3) [Multiplying numerator and denominator by √3]
h = (50√3) / 3
So, the height of the lighthouse is approximately 28.87 meters.
Therefore, the correct option is:
b) 25
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