Question

A flag post 10m long is fixed on top of a tower .from a point on horizontal ground the angel of elevation of the top and bottom of the flag post are 40° and 30 ° respectively.calculate the height of the tower

Answer 1

Given :

• Height of the flag post = 10 m.

• Angle of Elevation = ∠1 = 40°

• Angle of Elevation = ∠2 = 30°

To find :

The height of the tower.

Solution :

To find the Base of the figure :

According to the diagram , the length of DC = 10 m and the length of CB = x m.

Thus the length of DB is the sum of lengths of CB and DC .i.e,

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\boxed{\bf{DB = CB + DC}}$

$:\implies \bf{DB = x + 10}$

$\boxed{\therefore \bf{DB = (x + 10)\:m}}$

Hence the length of DB is (10 + x) m.

Hence by the above information , we can find the base of the figure in terms of variables.

We know that tan θ is P/B.

Where :

• P = Height

• B = Base

Let the base of the triangle be y m.

Using tan θ and substituting the values in it, we get :

$:\implies \bf{tan\:\theta = \dfrac{P}{B}}$

$:\implies \bf{tan40^{\circ} = \dfrac{10 + x}{y}}$

$:\implies \bf{0.839 = \dfrac{10 + x}{y}} \quad (\because tan40^{\circ} = 0.8390)$

$:\implies \bf{0.839y = 10 + x}$

$:\implies \bf{y = \dfrac{10 + x}{0.839}}$

$\boxed{\therefore \bf{y = \dfrac{10 + x}{0.839}}}$

Hence the value of base of the figure is $\bf{y = \dfrac{10 + x}{0.839}}$

To find the height of the tower :

We know that tan θ is P/B.

Where :

• P = Height

• B = Base

Using tan θ and substituting the values in it, we get :

$:\implies \bf{tan\:\theta = \dfrac{P}{B}}$

$:\implies \bf{tan30^{\circ} = \dfrac{x}{\dfrac{10 + x}{0.839}}}$

$:\implies \bf{tan30^{\circ} = \dfrac{x}{10 + x} \times 0.839}$

$:\implies \bf{tan30^{\circ} = \dfrac{0.839x}{10 + x}}$

$:\implies \bf{\dfrac{1}{\sqrt{3}} = \dfrac{0.839x}{10 + x}}$

$:\implies \bf{\dfrac{10 + x}{\sqrt{3}} = 0.839x}$

$:\implies \bf{\dfrac{10 + x}{1.732} = 0.839x}$

$:\implies \bf{10 + x = 0.839x \times 1.732}$

$:\implies \bf{10 + x = 1.5x}$

$:\implies \bf{10 = 1.5x - x}$

$:\implies \bf{10 = 0.5x}$

$:\implies \bf{\dfrac{10}{0.5} = x}$

$:\implies \bf{20 = x}$

$\boxed{\therefore \bf{x = 20\:m}}$

Hence the height of the tower is 20 m.