A person walking 20m towards a chimney in a horizontal line throu… - Vidyadip

MCQ

A person walking $20m$ towards a chimney in a horizontal line through its base observes that its angle of elevation changes from $30^{\circ}$ to $45^{\circ}$. The height of chimney is

✓
$\frac{20}{\sqrt{3}+1} m$
B
$\frac{20}{\sqrt{3}-1} m$
C
$20(\sqrt{3}-1) m$
D
None of these

✓

Answer

Correct option: A.

$\frac{20}{\sqrt{3}+1} m$

Suppose height of the chimney be $h$ metres.
Let $A$ and $B$ be the points of observation and $\text{BC}$ be $x m$. In $\triangle \text{ACD}$,

$\tan 30^{\circ}=\frac{CD}{AC}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{20+x}$
$\Rightarrow 20+x=h \sqrt{3}$
$\Rightarrow x=h \sqrt{3}-20$
Now, in $\triangle \text{DBC,} \tan 45^{\circ}=\frac{CD}{BC}$
$\Rightarrow 1=\frac{h}{x}$
$\Rightarrow x=h$
From $(i)$ and $(ii),$ we get
$h=h \sqrt{3}-20$
$\Rightarrow h \sqrt{3}-h=20$
$\therefore h=\frac{20}{\sqrt{3}-1} m$

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