Question

From the top of a building $20m$ high, the angle of elevation of the top of a monument is ${45}^o$ and the angle of depression of its foot is ${15}^o$. Find the height of the monument.

Answer 1

Consider

$AB$ as the building where $AB=20m$

$CD$ as the monument where $CD=x$ m.

Take the distance between building and the monument as $y$,

In right triangle $BCD$

$\tan \theta =\frac{CD}{BD}$

Substituting the values,

$\Rightarrow \tan {45}^o=\frac{x}{y}$

$\Rightarrow 1=\frac{x}{y}$

$\Rightarrow x=y...\left(1\right)$

In right triangle $ABD$,

$\tan {15}^o=\frac{AB}{BD}=\frac{20}{x}$

Substituting the values

$\Rightarrow 0.2679=\frac{20}{x}$

$\Rightarrow x=\frac{20}{0.2679}=74.65m$

Hence, the height of the monument is $74.65m$.