The beam of light along the shoreline when it is 1 km from P is moving at 62.83 km/min.
[tex]\frac{d\theta}{dt} =3\ rev/min=3*2\pi=6\pi\ rad/min\\\\\\tan\theta=\frac{x}{3} \\\\\frac{d}{dt} tan\theta=\frac{d}{dt}( \frac{x}{3} )\\\\sec^2\theta\frac{d\theta}{dt}=\frac{1}{3} \frac{dx}{dt} \\\\\frac{dx}{dt}=3sec^2\theta\frac{d\theta}{dt}\\\\\\At\ x=1\ km;tan\theta=\frac{x}{3} =\frac{1}{3} \\\\sec^2\theta=1+tan^2\theta=1+(\frac{1}{3})^2\\\\sec^2\theta=\frac{10}{9} \\\\\\\frac{dx}{dt}=3sec^2\theta\frac{d\theta}{dt}\\\\\frac{dx}{dt}=3*\frac{10}{9} *6\pi\\\\[/tex]
[tex]\frac{dx}{dt}=62.83\ km/min[/tex]
The beam of light along the shoreline when it is 1 km from P is moving at 62.83 km/min.
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