From a point P on the ground the angle of elevation of a 10 m tall building is \[{{30}^{\circ }}\], A flag is hoisted at the top of the building and the angle of elevation of the top of the flag-staff from P is \[{{45}^{\circ }}\]. Find the length of the flag-staff and the distance of the building from the point P. (Take \[\sqrt{3}=1.732\])

Hint: Consider the height of the building as AB and draw the angle of elevations at two different points and apply \[\tan \theta \] to the two right angled triangles and we will get two equations and then we have to compute the necessary quantities.

Complete step-by-step answer:



Distance of the building,

In \[\Delta BAP\]

\[\tan {{30}^{\circ }}=\dfrac{AB}{AP}\]

\[\dfrac{1}{\sqrt{3}}=\dfrac{10}{x}\]

\[x=10\sqrt{3}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

Distance of the building is \[x=10\sqrt{3}\].

Now the length of the flag is,

In \[\Delta PAD\]

\[\tan {{45}^{\circ }}=\dfrac{AD}{AP}\]

\[1=\dfrac{AD}{AP}\]

From the figure,

\[x=AD=DB+BA\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

Putting (1) in (2) we get,

\[10\sqrt{3}=h+10\]

\[h=10\sqrt{3}-10\]

\[=10\left( \sqrt{3}-1 \right)\]

\[h=10\left( 1.732-1 \right)\]

\[h=7.32m\].

Length of the flag is = \[h=7.32m\].

Note: The angle of elevation is the angle between the horizontal line from the observer and the line of sight to an object that is above the horizontal line. As the person moves from one point to another angle of elevation varies. If we move closer to the object the angle of elevation increases and vice versa.