question_answer

From a point P on the ground the angle of elevation of the top of a tower is \[30{}^\circ \] and that of the top of a flag staff fixed on the top of the tower, is \[60{}^\circ \]. If the length of the flag staff is 5 m, find the height of the tower.

Answer:

Let CB be the tower of x m and AC be the flag staff of 5 m.

Then, in \[\Delta \text{ }CPB\]

\[\tan \,30{}^\circ =\frac{x}{PB}\]

\[PB=\frac{x}{\tan \,30{}^\circ }=\sqrt{3}x\]                             ?(i)

In\[\Delta \text{ }APB\]

\[\tan \,60{}^\circ =\frac{5+x}{PB}\]

\[PB=\frac{5+x}{\sqrt{3}}\]                                          ?(ii)

From eq. (i) and (ii)

\[\sqrt{3}x=\frac{x+5}{\sqrt{3}}\]

\[\Rightarrow x=\frac{x+5}{3}\]

\[\Rightarrow 3x-x=5\]

\[\Rightarrow 2x=5\]

\[\Rightarrow x=5/2=2.5\]

\[\therefore \] Height of the tower is \[2.5\text{ m}\].