question_answer

A person standing on the bank of a river observes that the angle of elevation of the top of the tree on the opposite bank of the river is \[60{}^\circ \] and when he retires 40 m away from the tree, the angle of elevation becomes \[30{}^\circ \]. The breadth of the river is

A)  20m     

B)                                     30m

C)  40m          

D)                    60m

Correct Answer: A

Solution :

Let \[CD\text{ }\left( =h \right)\]be the height of the tree and \[BC(=x)\] be the width of the river.             In \[\Delta ABC,\,\,\tan {{60}^{0}}=\frac{CD}{BC}\]             \[\Rightarrow \,\,\sqrt{3}\,=\frac{h}{x}\Rightarrow \,h=x\sqrt{3}\]              ?(i)             In \[\Delta ACD,\,\tan {{30}^{0}}\,=\frac{CD}{AC}\]             \[\Rightarrow \]            \[\frac{1}{\sqrt{3}}\,=\frac{h}{40+x}\,\Rightarrow \,h\sqrt{3}\,=40+x\]             \[\Rightarrow \]            \[3x=40+x\] [from Eq. [a]]             \[\Rightarrow \]            \[x=20m\]             Hence, breadth of the river is 20 m.