Question

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a $20m$ high building are $45°$ and $60°$ respectively. Find the height of the tower.

Answer 1

Solve for height of the tower

Given,

1. Height of the building, $BC=20m$
2. The angles of elevation of the bottom and the top of a transmission tower are $45°$ and $60°$ respectively.

From the figure,

$$\begin{gathered} \tan \left(45°\right)=\frac{BC}{CD} \\ \Rightarrow 1=\frac{BC}{CD} \\ \Rightarrow BC=CD \\ \Rightarrow CD=20m \end{gathered}$$

Now,

$$\begin{gathered} \tan \left(60°\right)=\frac{AC}{CD} \\ \Rightarrow \sqrt{3}=\frac{AC}{CD} \\ \Rightarrow AC=\sqrt{3}CD=20\sqrt{3}m \\ \therefore AB=AC-BC=20\left(\sqrt{3}-1\right)=14.64m \end{gathered}$$

Hence, height of the tower is $14.64m$.