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 $PQ$ be the height of the building.
$\therefore PQ=10\mathrm{m}\text{ (given) }$
Let $AB$ be the height of the tower.
In $△PQA$,
$\tan 60^{\circ }=\frac{PQ}{PA}\Rightarrow \sqrt{3}=\frac{10}{PA}$
$\therefore PA=\frac{10}{\sqrt{3}}$
$\therefore QC=\frac{10}{\sqrt{3}}$

In $△QBC$,
$\tan 30^{\circ }=\frac{BC}{QC}\Rightarrow \frac{1}{\sqrt{3}}=\frac{BC}{\frac{10}{\sqrt{3}}}$
$\therefore BC=\frac{10}{3}$
From the figure,
$AB=AC+BC=10+\frac{10}{3}=\frac{40}{3}\mathrm{m}$