Answer
The beam of light is moving along the shoreline at a speed of $~~83.8~km/min$
Work Step by Step
Let $x$ be the distance from the light on the shore to point P.
Let $y$ be the distance from the lighthouse to point P.
Let $z$ be the distance from the lighthouse to the light on the shore.
$z^2 = x^2+y^2$
$z^2 = 1^2+3^2$
$z^2 = 10$
$z = \sqrt{10}~km$
Let $\theta$ be the angle between the beam of light and the line connecting the lighthouse and point P.
We can find $\frac{d\theta}{dt}$:
$\frac{d\theta}{dt} = \frac{(4)(2\pi)~rad}{min} = (8\pi)~rad/min$
We can find $\frac{dx}{dt}$:
$\frac{x}{y} = tan~\theta$
$\frac{1}{y}~\frac{dx}{dt}-\frac{x}{y^2}~\frac{dy}{dt} = (sec^2~\theta)~\frac{d\theta}{dt}$
$\frac{1}{3}~\frac{dx}{dt}-\frac{x}{y^2}~(0) = (sec^2~\theta)~\frac{d\theta}{dt}$
$\frac{dx}{dt} = (3)(sec^2~\theta)~\frac{d\theta}{dt}$
$\frac{dx}{dt} = (3)(\frac{\sqrt{10}}{3})^2~(8\pi)$
$\frac{dx}{dt} = \frac{80~\pi}{3}~km/min$
$\frac{dx}{dt} = 83.8~km/min$
The beam of light is moving along the shoreline at a speed of $~~83.8~km/min$
