The angle of elevation of a cloud from a point 200 meter above a water body is 30° and the angle of depression of its reflection in the water body is 60°. The height of the cloud in meter is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 200
Note: The given answer is as per the official NTA Answer Key. However, there seems to be a mistake in the question. The cloud is 400 m above the water body, which is not among the options; 200 m is its height above the point of observation, as explained below. NTA measured the cloud from where the observer stands, so 200 m of its height went missing.
The reflection C′ of the cloud C lies as far below the water surface as the cloud is above it. Let the observer be at A, 200 m above the water, and let the horizontal line through A meet the vertical through the cloud at M, with \(AM=d\) and \(CM=h\), the height of the cloud above A.
The cloud (angle of elevation 30°):
\(\displaystyle\tan30^{\circ}=\frac{h}{d}\), so \(d=\sqrt{3}\,h\) (1)
The reflection (angle of depression 60°):
The cloud is \(h+200\) m above the water, so C′ is \(h+200\) m below it and \(MC'=200+(h+200)=h+400\)
\(\displaystyle\tan60^{\circ}=\frac{h+400}{d}\), so \(d\sqrt{3}=h+400\) (2)
Putting (1) in (2):
\(3h=h+400\), so \(h=200\) m
The cloud is 200 m above the point of observation, which is the answer given; its height above the water surface is \(200+200=400\) m.