question_answer

A stone lies at the bottom of a stream. A boy wants to hit it with a stick. Taking aim the boy holds the stick in the air at an angle of \[45{}^\circ \]. At what distance from the stone will the stick hit the bottom. If the depth is 3; cm (given \[_{a}{{u}_{w}}\] 4/3) :

A)  12\[\sqrt{2}\] cm                           

B)  16 cm

C)  12 cm                                  

D)  8 cm

Correct Answer: C

Solution :

                Here,  \[\angle PQN={{45}^{o}}\] So,   \[PN=h\,\tan \,{{45}^{o}}=32m\]                 Now,   \[\mu =\frac{\sin i}{\sin r}\] So,          \[\sin r=\frac{sin\,i}{\mu }\] or            \[\sin r=\frac{\sin \,{{45}^{o}}}{(4/3)}=\frac{3}{4\sqrt{2}}\]                 \[\tan r=\frac{\sin r}{\cos r}\]                 \[=\frac{\sin \,r}{\sqrt{1-{{\sin }^{2}}r}}\]                 \[=\frac{3/4\sqrt{2}}{\sqrt{1-\frac{9}{32}}}=\frac{3}{\sqrt{23}}\] So,    ON= h tan r                 \[=32\times \frac{3}{\sqrt{23}}=20cm\]                 \[OP=PN-ON\]                 \[=32-20=12\text{ }cm\]