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Question
A man on the deck of a ship is 10 m above water level. He observes that the angle of elevation of the top of a cliff is 42° and the angle of depression of the base is 20°. Calculate the distance of the cliff from the ship and the height of the cliff.
Solution
Let the height of the cliff be h meters and the distance of the cliff from the ship be x meters.
In right-angled ΔQRS,
∴ QR = ST = 10 m
TQ = RS = x m
∴ tan 70° = `"RS"/"QR"`
⇒ `2.747 = x/(10 m)`
∴ x = 27.47 m
Hence, the distance of the cliff from the ship = 27.47 m
Again in right angled ΔPRS,
∴ tan 42° = `"PR"/"RS"`
⇒ `0.9004 = "PR"/27.47`
⇒ PR = [ 0.9004 x 27.47 ] m
⇒ PR = 24.73 m
∴ PQ = PR + RQ
PQ = [ 24.73 + 10 ] m
PQ = 34.73 m
Hence the height of the cliff = 34.73 m.
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