Question

The angle of elevation of the top of a building from the foot of the tower is \[{{30}^{0}}\] and the angle of elevation of the top of the tower from the foot of the building is \[{{60}^{0}}\]. if the tower is 50m high. find the height of the building.

Answer 1

Hint: To solve the question, we have to represent the given information in diagram format which will ease the process of analysing the given information. Then applying the properties of trigonometric angles of triangles to the formed triangles to calculate the height of the building.

Complete step by step answer:
Let AB and CD be the height of the tower and the height of the building respectively.
Given that the height of the tower = 50 m
Thus, AB = 50 m ….. (1)
Let BC be the distance between the foot of the tower and the foot of the building.
Given that the angle of elevation given from the foot of tower and from the top of tower to the top of the building are \[{{30}^{0}},{{60}^{0}}\] respectively.

We know the property that \[\tan \alpha \]is equal to the ratio of the opposite side to the adjacent side of the triangle.
Where \[\alpha \]is the angle between the hypotenuse and the adjacent side of the triangle.
Thus, by applying the property to the given angle of elevations, we get
\[\begin{align}
  & \tan {{30}^{0}}=\dfrac{CD}{BC} \\
 & \tan {{60}^{0}}=\dfrac{AB}{BC} \\
\end{align}\]
We know that \[\tan {{30}^{0}}=\dfrac{1}{\sqrt{3}},\tan {{60}^{0}}=\sqrt{3}\]
By applying the above values and equation (1) we get,
\[\begin{align}
  & \dfrac{1}{\sqrt{3}}=\dfrac{CD}{BC} \\
 & \Rightarrow BC=\sqrt{3}CD \\
\end{align}\]
\[\begin{align}
  & \sqrt{3}=\dfrac{50}{BC} \\
 & \Rightarrow BC=\dfrac{50}{\sqrt{3}} \\
\end{align}\]
By equating the above two equations, we get
\[\begin{align}
  & \sqrt{3}CD=\dfrac{50}{\sqrt{3}} \\
 & CD=\dfrac{50}{\sqrt{3}\times \sqrt{3}} \\
 & \Rightarrow CD=\dfrac{50}{3}m \\
\end{align}\]
Thus, the value of CD is equal to \[\dfrac{50}{3}=16.67m\]
Thus, the height of the building is equal to 16.67 m.
Note: The possibility of mistake is not representing the given information in diagram format which eases the process of analysing the given information. The other possibility of mistake can be not applying the properties of trigonometric angles of triangles which ease the procedure of solving. The alternative way of solving is to apply the ASA (Angle Side Angle) property to the two triangles formed in the diagrammatic representation of the given data. Thus, we can calculate the height of the building.