Let x=distance from point on shore nearest lighthouse to point where beams hits the shore.
We want \(\displaystyle \L\\\frac{dx}{dt}\) when x=1
We have \(\displaystyle \L\\\frac{d{\theta}}{dt}=(4\frac{rev}{min})(\frac{2{\pi}rad}{1rev})=8{\pi}\;\ \frac{rad}{min}\)
\(\displaystyle \L\\x=3tan({\theta})\)
\(\displaystyle \L\\\frac{dx}{dt}=3sec^{2}({\theta})\frac{d{\theta}}{dt}\)
When x=1, \(\displaystyle \L\\tan({\theta})=\frac{1}{3}\)
Then, \(\displaystyle \L\\sec({\theta})=\frac{\sqrt{10}}{3}\)
You have the info, find
\(\displaystyle \L\\\frac{dx}{dt}=3sec^{2}({\theta})\frac{d{\theta}}{dt}\)
