A man on the deck of a ship, 16 m above water level, observes that the angles of elevation and depression respectively of the top and bottom of a cliff are 60° and 30°. Calculate the distance of the cliff from the ship and height of the cliff. [Take √3 = 1.732.]

In the above figure, let B be the position of the man and AE be the cliff. We are given that the position of the man is 16 m above water level. Draw a line BD to a point D on the water level. So, BD = 16 m. Now, we are given that the angle of elevation of the top of the tower from the position of the man is 60°. Join A and B. Also, draw a line BC on to the line AE parallel to the water level. Then we get a right angled triangle ABC with right angle at B and ABC = 60°. Also, the angle of depression of the bottom of the tower from the position of the man is 30°. So, CBE = 30°. Joining B and E we get a right angled triangle BDE. And, BED = CBE = 30°. We need to find the distance of the ship from the cliff, that is DE and the height of the cliff AE.

We first find DE from the ∆BDE by using the trigonometric ratio tan.

In ∆BDE,

or,

or,

Now, BC = DE = 16√3 m.

From ∆ABC,

or,

or,

Also, CE = BD = 16 m.

Hence, the height of the cliff is AE = AC + CE = 48 + 16 = 64 m.

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