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Mathematics

A flagstaff of height (15)\Big(\dfrac{1}{5}\Big) of the height of a tower is mounted on the top of the tower. If the angle of elevation of the top of the flagstaff as seen from the ground is 45° and the angle of elevation of the top of the tower as seen from the same place is θ, then the value of tan θ is:

  1. 45\dfrac{4}{5}

  2. 56\dfrac{5}{6}

  3. 65\dfrac{6}{5}

  4. 536\dfrac{5\sqrt{3}}{6}

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Answer

A flagstaff of height of the height of a tower is mounted on the top of the tower. If the angle of elevation of the top of the flagstaff as seen from the ground is 45° and the angle of elevation of the top of the tower as seen from the same place is θ, then the value of tan θ is: Volume And Surface Area of solid RSA Mathematics Solutions ICSE Class 10.

Let the height of the tower (BC) be H and AB be the height of the flag staff = H5\dfrac{H}{5}

Total height (AC) = H + H5=6H5\dfrac{H}{5} = \dfrac{6H}{5}

Let P be the point of observation at distance d from tower.

In triangle PCA,

⇒tan⁡45∘=ACCD⇒1=6H5d⇒d=6H5.\Rightarrow \tan 45^{\circ} = \dfrac{AC}{CD} \\[1em] \Rightarrow 1 = \dfrac{\dfrac{6H}{5}}{d} \\[1em] \Rightarrow d = \dfrac{6H}{5}.

In triangle PCB,

⇒tan⁡θ=Hd⇒tan⁡θ=H6H5⇒tan⁡θ=165⇒tan⁡θ=56.\Rightarrow \tan \theta = \dfrac{H}{d} \\[1em] \Rightarrow \tan \theta = \dfrac{H}{\dfrac{6H}{5}} \\[1em] \Rightarrow \tan \theta = \dfrac{1}{\dfrac{6}{5}} \\[1em] \Rightarrow \tan \theta = \dfrac{5}{6}.

Hence, option 2 is the correct option.

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