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A person standing on the bank of a river observes that the angle subtended by a tree on the opposite of bank is $60^{\circ}$. When he retires $40 \mathrm{~m}$. from the bank, he finds the angle to be $30^{\circ} .$ What is the breadth of the river?
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$20 \mathrm{~m}$
$\operatorname{In} \Delta \mathrm{ABC}, \tan 60^{\circ}=\frac{\mathrm{h}}{\mathrm{x}}$
$\Rightarrow \sqrt{3}=\frac{\mathrm{h}}{\mathrm{x}} \Rightarrow \mathrm{h}=\sqrt{3} \mathrm{x}$
$\operatorname{In} \Delta \mathrm{ABD}, \tan 30^{\circ}=\frac{\mathrm{h}}{\mathrm{x}+40}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{h}}{\mathrm{x}+40} \Rightarrow \mathrm{x}+40=\sqrt{3} \mathrm{~h}$
Putting value of h from equation (1), we get $x+40=3 x$
$\mathrm{x}=20 \mathrm{~m}$
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