Question

A flagpole stands on a building of height $450{\text{ }}ft$ and an observer on level ground is $300{\text{ }}ft$ from the base of the building . The angle of elevation of the bottom of the flagpole is $30^\circ$ and the height of the flagpole is $50{\text{ }}ft$ . If $\theta$ is the angle of elevation of the top of the flagpole , then $\tan \theta$ is
$\left( 1 \right)$ $\dfrac{4}{{3\sqrt 3 }}$
$\left( 2 \right)$ $\dfrac{{\sqrt 3 }}{2}$
$\left( 3 \right)$ $\dfrac{9}{2}$
$\left( 4 \right)$ $\dfrac{{\sqrt 3 }}{5}$
$\left( 5 \right)$ $\dfrac{{4\sqrt 3 + 1}}{6}$

Answer 1

We have to find the value of $\tan \theta$ in the given problem . We solve this question using the concept of applications of trigonometry . We should have the knowledge of the basic trigonometric functions and their values . We would construct a diagram for a reference and using the various relations of trigonometry and using the values of trigonometric functions we will find the value of $\tan \theta$.

Complete answer: Given :
A flagpole stands on a building of height $450{\text{ }}ft$ and an observer on level ground is $300{\text{ }}ft$ from the base of the building . The angle of elevation of the bottom of the flagpole is $30^\circ$ and the height of the flagpole is $50{\text{ }}ft$ . $\theta$ is the angle of elevation of the top of the flagpole .
Construct : According to the question , we construct a diagram as shown .

Now ,
In triangle $DCE$ ,
$\tan {30^ \circ } = \dfrac{{DE}}{{DC}}$
[As we know that $tan{\text{ }}x{\text{ }} = {\text{ }}\dfrac{{perpendicular}}{{base}}$]
As , we know that $\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$
We get ,
$\dfrac{1}{{\sqrt 3 }} = \dfrac{{DE}}{{DC}}$
$CD = \sqrt 3 \times DE$
And , $DE{\text{ }} = {\text{ }}150$
So ,
$CD = 150\sqrt 3 - - - (1)$
In triangle $DCF$ ,
$\tan \theta = \dfrac{{FD}}{{CD}}$
$\left[ {FD{\text{ }} = {\text{ }}150{\text{ }} + {\text{ }}50{\text{ }} = {\text{ }}200} \right]$
$\tan \theta = \dfrac{{200}}{{150\sqrt 3 }}$
Cancelling the terms , we get
$\tan \theta = \dfrac{4}{{3\sqrt 3 }}$
Thus , the value of $\tan \theta$ is $\dfrac{4}{{3\sqrt 3 }}$ .
Hence , the correct option is $\left( 1 \right)$ .

Note:
Applications of trigonometry : It is the property or the concept of trigonometry which can be used in our daily life to either find the length or the angle of elevations of a body , building , the length of the formation of shadow of an object due to light . We can also compute the value or the measurement for the depth in the river of an object above it . This can also help us in finding the speed of an object moving in a direction which is seen by a person sitting on a height .