Question

The horizontal distance between two towers is $60\mathrm{m}$ and the angle of depression of the top of the first tower as seen from the top of the second is $30°.$ If the height of the second tower is $90\mathrm{m}$, then the height of the first tower is

• A. $90\mathrm{m}$
• B. $\left(150-60\sqrt{3}\right)\mathrm{m}$
• C. $\left(150-20\sqrt{3}\right)\mathrm{m}$
• D. None of these

Answer 1

The correct option is D

None of these

The explanation for the correct answer.

Find the height of the other tower.

Given that: Height of the second tower is $90\mathrm{m}$ and the distance between the towers is $60\mathrm{m}.$

From the figure, $\mathrm{DE}$ and $\mathrm{AB}$ are the first and second towers respectively.

$\angle \mathrm{C}=90°\mathrm{and}\mathrm{AB}=90\mathrm{m}$

Therefore in triangle $\mathrm{BCE}$

$$\begin{gathered} \Rightarrow \tan 30°=\frac{\mathrm{BC}}{\mathrm{CE}}\left[\therefore \mathrm{tan\theta }=\frac{\mathrm{perpendicular}}{\mathrm{base}}\right] \\ \Rightarrow \tan 30°=\frac{90-h}{60} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{90-\mathrm{h}}{60}\left[\therefore \mathrm{BC}=90-\mathrm{h}\right] \\ \Rightarrow 60\sqrt{3}=90-h \\ \Rightarrow \mathrm{h}=90-60\sqrt{3} \end{gathered}$$

Hence option $\left(d\right)$ is the correct answer.