Question

A statue, $1.6m$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60°$ and from the same point the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.

Answer 1

Let $AB$ be the height of the statue.

$D$ is the point on the ground from where the elevation is taken.

To find the eight of pedestal $=BC=AC-AB$

From the figure AB $=statue=1.6m$

$BC=pedestal=?$

In $∆ACD$

$\frac{P}{B}=\frac{AC}{CD}=\tan 60°$

$1.6+\frac{BC}{CD}=\sqrt{3}$

$\sqrt{3}CD=1.6+BC$

$CD=1.6+\frac{BC}{\sqrt{3}}---------------\left\{1\right\}$

In $∆BCD,$

$$\begin{gathered} \frac{BC}{CD}=\tan 45° \\ \frac{BC}{CD}=1 \end{gathered}$$

$CD=BC$

From (1)

$\frac{1.6+BC}{\sqrt{3}}=\frac{BC}{1}$

$\sqrt{3}BC=1.6+BC$

$1.732BC–1BC=1.6$

$0.732BC=1.6$

$BC=\frac{1.6}{0.732}$

$BC=\frac{16}{10}\times \frac{100}{732}=\frac{1600}{732}$

$BC=2.18m$

Hence, the height of the pedestal is$2.18m$.