Q.20. A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60$°$ to 45$°$ in 2 minutes. Find the speed of the boat in m/h.
 

Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As :

Here , Height of cliff =  AC = 150 m 

And boat's initial position is at ' D '  and position of boat after 2 minutes is at ' B ' .

We know : $\tan \mathrm{\theta } = \frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$ , So

In triangle ABC we get :

$$\begin{gathered} \tan \left(\angle \mathrm{ABC}\right) = \frac{\mathrm{AC}}{\mathrm{AB}} \\ \Rightarrow \tan 45° = \frac{150}{\mathrm{AB}} \\ \Rightarrow 1 = \frac{150}{\mathrm{AB}} ( \mathrm{We} \mathrm{know} : \tan 45° =1 ) \\ \mathbf{\Rightarrow }\mathbf{AB}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{150}\mathbf{ } \end{gathered}$$


And in triangle ADC we get :

$$\begin{gathered} \tan \left(\angle \mathrm{ADC}\right) = \frac{\mathrm{AC}}{\mathrm{AD}} \\ \Rightarrow \tan 60° = \frac{150}{\mathrm{AD}} \\ \Rightarrow \sqrt{3} = \frac{150}{\mathrm{AD}} ( \mathrm{We} \mathrm{know} : \tan 60° =\sqrt{3} ) \\ \Rightarrow \mathrm{AD} = \frac{150}{\sqrt{3}} \\ \Rightarrow \mathrm{AD} = \frac{150}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \\ \Rightarrow \mathrm{AD} = \frac{150\sqrt{3}}{ 3} \\ \mathbf{\Rightarrow }\mathbf{AD}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{50}\sqrt{\mathbf{3}}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{50}\mathbf{ }\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{732}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{86}\mathbf{.}\mathbf{6} \end{gathered}$$
So,

Distance travelled by boat in 2 minutes = AB - AD =  150 - 86.6 = 63.4 m

And time taken to cover distance 63.4 m = 2 minutes = $\left(2 \times \frac{1}{60}\right)\mathrm{hr} = \left(\frac{1}{30}\right)\mathrm{hr}$

We know :  Speed = $\frac{\mathrm{Distance}}{\mathrm{Time}}$

Therefore,

Speed of boat = $\frac{63.4}{{\displaystyle \frac{1}{30}}} = 63.4 \times 30 = \mathbf{1902}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{hr}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)}$


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