Question

From a window, $20$m above the ground the angle of elevation of the top of a tower is $x^{\circ }$ where $\tan x^{\circ }=\frac{5}{2}$ and the angle of depression of the foot of the tower is $y^{\circ }$ where $\tan y^{\circ }=\frac{1}{4}$. Calculate the height of the tower in meters.

Answer 1

Let $A$ be the window, $CE$ be the height of the tower.

Here $AB=20$m

$\Rightarrow CD=20$m

$\angle DAE=x^{\circ }$ and $\angle DAC=y^{\circ }$

$\Rightarrow \angle ACB=y^{\circ }$(alternate interior angles)

In right-angled triangle$△ADE$

$\tan x^{\circ }=\frac{DE}{AD}$

$\Rightarrow \frac{5}{2}=\frac{DE}{AD}$

$\Rightarrow DE=\frac{5}{2}AD$

In right-angled triangle$△ABC$

$\tan y^{\circ }=\frac{AB}{BC}$

$\Rightarrow \frac{1}{4}=\frac{20}{BC}$

$\Rightarrow BC=80$

Here $DE=\frac{5}{2}AD=\frac{5}{2}BC=\frac{5}{2}\times 80=200$ since $AD=BC$

Height of the tower$=CD+DE=20+200=220$m