Question

The angle of elevation of the top of a tower at a distance 500 meters from the foot is \[30^\circ \]. The height of the tower is
A.\[250\sqrt 3 m\]
B.\[\dfrac{{500}}{{\sqrt 3 }}m\]
C.\[500\sqrt 3 m\]
D.\[250m\]

Answer 1

Hint: We will first consider the given data that the angle of elevation is \[30^\circ \] and at the distance of 500 m from the foot of the tower. We need to find the height of the tower by constructing a triangle and mark the given data in the figure and using the figure we will find the height by finding the value of \[\tan 30^\circ \] and thus will get the desired answer.

Complete step-by-step answer:
We will first make the figure and represent the given data through it that is the angle of elevation is \[30^\circ \] and the distance from the foot of the tower is 500m.
We have to determine the value of height.
Thus, we get the figure as:

Now, we will use the trigonometric identity which is \[\tan \left( \alpha \right) = \dfrac{P}{B}\] where \[P\] shows the perpendicular and \[B\] shows the base and \[\alpha \] is the angle of elevation. As we need to find the height so, we have taken the \[\tan \] function,
Thus, we get,
\[ \Rightarrow \tan \left( {30} \right) = \dfrac{h}{{500}}\]
Where \[h\] is the perpendicular and the base is shown by 500 m.
Now, as we know that the value of \[\tan \left( {30} \right) = \dfrac{1}{{\sqrt 3 }}\], we will substitute the value in the above expression,
Thus, we get,
\[
   \Rightarrow \dfrac{1}{{\sqrt 3 }} = \dfrac{h}{{500}} \\
   \Rightarrow h = \dfrac{{500}}{{\sqrt 3 }}m \\
 \]
Thus, we can conclude that the height of the tower is \[\dfrac{{500}}{{\sqrt 3 }}m\].
Hence, option B is correct.
Note: We have used the trigonometric identity to find the height as in such questions, we have to remember these concepts. We have used the \[\tan \] function because it has base and perpendicular in its property and there is no use of hypotenuse as we have to find the perpendicular that is height and have the value of base as 500 m. the figure makes the concept more clear and understandable so draw the figure before starting the solution.