Three villagers A, B and C can see each other using telescope acr… - Vidyadip

Question

Three villagers $A, B$ and $C$ can see each other using telescope across a valley. The horizontal distance between $A$ and $B$ is $8 km$ and the horizontal distance between $B$ and $C$ is $12 km$. The angle of depression of $B$ from $A$ is $20^{\circ}$ and the angle of elevation of $C$ from $B$ is $30^{\circ}$. Calculate the vertical height between $A$ and $B .\left(\tan 20^{\circ}=0.3640, \sqrt{3}=1.732\right)$

✓

Answer

Let AD is the vertical height between A and B
In the right ∆ABD

$
\begin{aligned}
& \tan 20^{\circ}=\frac{A D}{B D} \\
& 0.3640=\frac{A D}{8} \\
& A D=0.3640 \times 8=2.912 km \\
& \therefore A D=2.91 km
\end{aligned}
$
$C E$ is the vertical height between $C$ and $B$ In the right $\triangle BCE , \tan 30^{\circ}=\frac{ CE }{ BE }$
$
\begin{aligned}
& \frac{1}{\sqrt{3}}=\frac{ CE }{12} \\
& \Rightarrow \sqrt{3} CE =12
\end{aligned}
$

$
\begin{aligned}
& \text { CE }=\frac{12}{\sqrt{3}} \\
& =\frac{12 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} \\
& =\frac{12 \times \sqrt{3}}{3} \\
& =4 \sqrt{3} \\
& =4 \times 1.732 \\
& =6.928 \\
& =6.93 km
\end{aligned}
$
The vertical height between $A$ and $B=2.91 km$

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