Math, asked by Anonymous, 2 months ago

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A Tower of height 5 metre and flagstaff on the top of a tower subtends equal angle at the point at ground distance 13 m from the base of Tower find the height of flagstaff .

( In m upto 2 decimal places )

Answers

Answered by SparklingBoy

187

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▪ Given :-

A Tower of height 5 metre and flagstaff on the top of a tower subtends equal angle at the point at ground distance 13 m from the base of Tower.

That is :

Height of Tower = 5 meter

Distance of Point from Tower = 13 meter

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▪ To Find :-

The Height of Flagstaff.

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▪ Solution :-

In The Attached Figure :

AB = Tower

BD = Flagstaff

Angle Subtended = θ

Let

Height of Flagstaff = x meter

According to the Figure :

$\Large\bigstar\:\: \underline{\pmb{ \mathcal{In \:\:\triangle\:\:DBC}} } : -$

$\pink{ \large\boxed{ \boxed{\mathtt{ \tan \theta = \dfrac{5}{13} }}}}\:\:\:\:---(1)$

Now ,

$\Large\bigstar\:\:\underline{ \pmb{\mathcal{In \:\:\triangle\:\:ABC}}} : -$

$\large \mathtt{ \tan2 \theta = \dfrac{5 + x}{13} }\\\\$

$\large \mathtt{ \dfrac{2 \tan \theta}{1 - \tan {}^{2} \theta} = \frac{5 + x}{13} } \\ \\$

Putting Value of tan θ from (1) :

$\large : \longmapsto \mathtt{ \dfrac{2 \times \dfrac{5}{13} }{1 - \dfrac{25}{169} } = \dfrac{5 + x}{13} } \\ \\ \large : \longmapsto \mathtt{ \dfrac{ \dfrac{10}{13} }{ \dfrac{169 - 25}{169} } = \dfrac{5 + x}{13} } \\ \\ \large : \longmapsto \mathtt{ \dfrac{ \dfrac{10}{13} }{ \dfrac{144}{169} } = \dfrac{5 + x}{13} } \\ \\ \large : \longmapsto \mathtt{\dfrac{130}{144} = \dfrac{5 + x}{13} } \\ \\\large : \longmapsto \mathtt{1690 = 720 + 144x} \\ \\ \large : \longmapsto \mathtt{144x = 970}\\ \\ \LARGE \purple{ :\longmapsto \underline {\boxed{{\bf x = 6.73m } }}}$