question_answer

    A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is \[60{}^\circ \]. When he retreats 20 ft from the bank, he finds the angle to be\[30{}^\circ \]. The breadth of the river in feet is

A)  15                                         

B)  \[15\sqrt{3}\]

C)  \[10\sqrt{3}\]                                  

D) \[10\]

Correct Answer: D

Solution :

                Let AB be the tree of height h feet and breadth of river be\[x\]feet. From \[\Delta ABC,\]                 \[\tan 60{}^\circ =\frac{h}{x}\] \[\Rightarrow \]               \[h=\sqrt{3}x\]                 ?..(i) and in \[\Delta ABD,\]                 \[\tan 30{}^\circ =\frac{h}{x+20}\] \[\Rightarrow \]               \[\frac{1}{\sqrt{3}}=\frac{h}{x+20}\] \[\Rightarrow \]               \[x+20=\sqrt{3}h\] \[\Rightarrow \]         \[x+20=3x\]                    [from Eq. (i)] \[\therefore \]  \[2x=20\] \[\Rightarrow \]               \[x=10\]               \[\therefore \]Breadth of river is 10 ft.