Question

The shadow of a tower is found to be $60m$ shorter when the Sun's altitude changes from $30°$ to $60°$.

The height of the tower from the ground is approximately

• A. $62m$
• B. $301m$
• C. $101m$
• D. $75m$
• E. $52m$

Answer 1

The correct option is E

$52m$

Explanation for correct option:

Given that the shadow of a tower is found to be $60m$ shorter when the Sun's altitude changes from $30°$ to $60°$

Let the height of the tower be $AB$

According to the question,

$$\begin{gathered} \angle ACB=60° \\ \angle ADB=30° \end{gathered}$$

When the angle is $\angle ACB=60°$ then the length of the shadow is $BCm$

When the angle is $\angle ADB=30°$ then the length of the shadow is $BD=CB+60m$

In $△ABC$,

$$\begin{gathered} \tan \left(60°\right)=\frac{AB}{CB} \\ \Rightarrow \sqrt{3}=\frac{AB}{CB} \\ \Rightarrow CB\sqrt{3}=AB \end{gathered}$$

In $△ABD$,

$$\begin{gathered} \tan \left(30°\right)=\frac{AB}{DB} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{CB\sqrt{3}}{CB+60} \\ \Rightarrow 3CB=CB+60 \\ \Rightarrow 2CB=60 \\ \Rightarrow CB=30m \end{gathered}$$

Therefore height of the tower $AB=CB\sqrt{3}$

$$\begin{gathered} =30\sqrt{3} \\ =51.96m \\ \approx 52m \end{gathered}$$

Hence, option(E) i.e. $52m$ is correct