Question

A flagpole stands on a building of height $450\mathrm{ft}$ and an observer on level ground is $300\mathrm{ft}$ from the base of the building. The angle of elevation of the bottom of the flagpole is $30°$ and the height of the flagpole is$50\mathrm{ft}$. If $\mathrm{\theta }$ is the angle of elevation of the top of the flagpole, then $\mathrm{tan\theta }$ is

• A. $\frac{4}{3\sqrt{3}}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{9}{2}$
• D. $\frac{\sqrt{3}}{5}$
• E. $\frac{(4\sqrt{3}+1)}{6}$

Answer 1

The correct option is A

$\frac{4}{3\sqrt{3}}$

Explanation for correct option

Step1.Draw the diagram

The height of the building is $\mathrm{ST}=450\mathrm{ft}$, Height of the flagpole is $\mathrm{PS}=50\mathrm{ft}$, $Q$ is the observation point.

The angle of elevation of the bottom of the flagpole is $30°$

The angle of elevation of the top of the flagpole is$\mathrm{\theta }$.

Let the distance between man and building be $\mathrm{RQ}=\text{'}\mathrm{a}\text{'}$

Step2. Find the value of$\mathbf{\text{'}}\mathbf{a}\mathbf{\text{'}}$.

In$∆\mathrm{SRQ}$, $$\begin{gathered} \tan 30°=\frac{\mathrm{SR}}{\mathrm{RQ}}[\therefore \mathrm{tan\theta }=\frac{perpendicular}{base}] \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{150}{\mathrm{a}} \\ \Rightarrow \mathrm{a}=150\sqrt{3} \end{gathered}$$

Step3. Find the value of $\mathrm{tan\theta }$

From the diagram we have,

$$\begin{gathered} \mathrm{PR}=\mathrm{PS}+\mathrm{SR} \\ =50+150 \\ =200\mathrm{ft} \end{gathered}$$

In$∆\mathrm{PQR},$

$\begin{array}{rcl}\mathrm{tan\theta }& =& \frac{\mathrm{PR}}{\mathrm{RQ}}\\ & \Rightarrow & \mathrm{tan\theta }=\frac{200}{150\sqrt{3}}\\ & \Rightarrow & \mathrm{tan\theta }=\frac{4}{3\sqrt{3}}\end{array}$

Hence, Option (A) is the correct answer.