Question

From the top of a building, the angles of elevation and depression of top and bottom of a tower are $30^{\circ}$ and $60^{\circ}$ respectively. If the height of the building is $10 \mathrm{~m}$, then find the height of the tower.

• A. $\frac{40}{3} m$
• B. $10 \sqrt{3} \mathrm{~m}$
• C. $30 \mathrm{~m}$
• D. $40 \mathrm{~m}$

Answer 1

Answer: (A)

Let $P Q$ be the height of the building.
$\therefore P Q=10 \mathrm{~m} \text { (given) }$
Let $A B$ be the height of the tower.
In $\triangle P Q A$,
$ \tan 60^{\circ}=\frac{P Q}{P A} \Rightarrow \sqrt{3}=\frac{10}{P A} $
$ \therefore P A=\frac{10}{\sqrt{3}} $
$\therefore Q C=\frac{10}{\sqrt{3}}$

In $\triangle Q B C$,
$ \tan 30^{\circ}=\frac{B C}{Q C} \Rightarrow \frac{1}{\sqrt{3}}=\frac{B C}{\frac{10}{\sqrt{3}}} $
$\therefore B C=\frac{10}{3}$
From the figure,
$A B=A C+B C=10+\frac{10}{3}=\frac{40}{3} \mathrm{~m}$