Question

From the top of a ciff, 60 m high, the angles
of depression of this top and bottom of a tower
are observed to be 30° and 60° - find
the height
of the tower​

Answer 1

Given:-

• Height of cliff = 60 m

• The angles of depression of this top and bottom of a tower are observed to be 30° and 60°

To find:-

• Height of the tower

Solution:-

Let AC be the tower and BE be the cliff.

Let the height of the tower be h m.

The angle of depression of the top C and bottom A of the tower, observed from top of cliff be 30° and 60° respectively.

★ In ∆ CDE, we have

$\dashrightarrow\sf \tan{30^\circ} = \dfrac{DE}{CD} \\\\\\ \dashrightarrow\sf \dfrac{1}{ \sqrt{3}} = \dfrac{60 - h}{CD} \qquad\bigg\lgroup\bf DE = BE - BD = BE - AC\bigg\rgroup \\\\\\ \dashrighgarrow\sf\dashrightarrow CD = \sqrt{3}(60 - h)\;\;\;\;\;-(i)$

★ In ∆ ABE, we have

$\dashrightarrow\sf \tan{60^\circ} = \dfrac{BE}{AB} \\\\\\ \dashrightarrow\sf \sqrt{3} = \dfrac{60}{CD} \qquad\bigg\lgroup\bf AB = CD\bigg\rgroup \\\\\\ \dashrighgarrow\sf\dashrightarrow CD = \dfrac{60}{ \sqrt{3}}\;\;\;\;\;-(ii)$

★ From eq. (i) and (ii), we get

$\sf \sqrt{3}(60 - h) = \dfrac{60}{ \sqrt{3}} \\ \\ \dashrightarrow\sf 3(60 - h) = 60 \\ \\ \dashrightarrow\sf 180 - 3h = 60 \\ \\ \dashrightarrow\sf 3h = 120 \\ \\ \dashrightarrow{\bf{\red{h = 40}}}$

Hence, The height of tower is 40m.

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