Question

A statue, $1.6\mathrm{m}$ tall, stands on the top of the pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60°$ and from the same point, the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.

• A. height of the pedestal $=0.2(\sqrt{7}–1)\mathrm{m}$
• B. height of the pedestal $=0.8(\sqrt{3}+1)\mathrm{m}$
• C. height of the pedestal $=1.6(\sqrt{5}+1)\mathrm{m}$
• D. height of the pedestal $=1.9(\sqrt{11}-1)\mathrm{m}$

Answer 1

The correct option is B

height of the pedestal $=0.8(\sqrt{3}+1)\mathrm{m}$

The explanation for the correct option.

According to the question, a diagram is drawn that is shown below.

In the above figure, $AF$ represent the statue whose height is $1.6\mathrm{m}$ and $FC$ represent the height of the pedestal, let say $h$.

In $∆FBC$,

$$\begin{gathered} \therefore \tan 45°=\frac{h}{m} \\ \Rightarrow 1=\frac{h}{m} \\ \Rightarrow m=h...\left(1\right) \end{gathered}$$

In $∆ABC$,

$$\begin{gathered} \therefore \tan 60°=\frac{AC}{BC} \\ \Rightarrow \sqrt{3}=\frac{1.6+h}{m} \\ \Rightarrow \sqrt{3}m=1.6+h \\ \Rightarrow \sqrt{3}h=1.6+h\left[\mathrm{From}\mathrm{eq}.1\right] \\ \Rightarrow h(\sqrt{3}-1)=1.6 \\ \Rightarrow h=\frac{1.6}{\sqrt{3}-1} \\ \Rightarrow h=\frac{1.6}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}\left[\mathrm{Rationalization}\right] \\ \Rightarrow h=\frac{1.6(\sqrt{3}+1)}{2} \\ \Rightarrow h=0.8(\sqrt{3}+1)m \end{gathered}$$

Thus, the height of the pedestal is $0.8(\sqrt{3}+1)\mathrm{m}$.

Hence, the option $B$ is correct.