From the top of a building 30 m high, the top and bottom of a tower are observed to have angles of depression 30º and 45º respectively. Find the height of the tower - Mathematics

Sum

From the top of a building 30 m high, the top and bottom of a tower are observed to have angles of depression 30º and 45º respectively. Find the height of the tower

Solution

Let AB be the building and CD be the tower.

Then, AB = 30 m. Let DC = x.

Draw DE ⊥ AB. Then AE = CD = x.

∴ BE = (30 – x) m.

\text{Now, }\frac{AC}{AB}=\text{cot 45}^\text{o}=1

\Rightarrow \frac{AC}{30}=1\Rightarrow AC=30 m

∴ DE = AC = 30 m.

\frac{BE}{DE}=\text{tan }30^\text{o}=\frac{1}{\sqrt{3}}

\Rightarrow\frac{BE}{30}=\frac{1}{\sqrt{3}}

\Rightarrow BE=\frac{30}{\sqrt{3}}

\therefore CD=AE=AB-BE=( 30-\frac{30}{\sqrt{3}} )

=30( 1-\frac{1}{\sqrt{3}})m

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