Mathematics
From the top of a cliff, the angles of depression of the top and bottom of a tower are observed to be 45° and 60° respectively. If the height of the tower is 20 m, find :
(i) the height of the cliff.
(ii) the distance between the cliff and the tower.
Heights & Distances
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Answer
(i) Let AB be the cliff and CD be the tower.
From figure,
∠ACE = ∠FAC = 45° (Alternate angles are equal)
∠ADB = ∠FAD = 60° (Alternate angles are equal)
Let BD = x.
From figure,
EC = BD = x meters.
EB = CD = 20 meters.
In △ AEC,
⇒ tan 45° =
⇒ 1 =
⇒ AE = x meters.
In △ ABD,
From figure,
Height of cliff (AB) = AE + EB
= x + 20
= 27.32 + 20
= 47.32 meters.
Hence, the height of cliff = 47.32 meters.
(ii) From figure,
Distance between cliff and tower (BD) = x meters = 27.32 meters.
Hence, distance between cliff and tower = 27.32 meters.
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