This problem describes a rotating lighthouse with a beam of light that sweeps across the shore. The lighthouse is 600 m from the shore, and the beam sweeps across the point $Q$ (which is 300 m from point $P$, the point on the shore closest to the lighthouse).

To calculate the speed of the beam at $Q$, we need to relate the angular velocity of the beam to the linear speed along the shore.

Steps:

1. Angular velocity: The beam revolves 3 times per minute. This means the angular velocity $\omega$ is: $\omega = 3 \times 2\pi \, \text{radians per minute} = 6\pi \, \text{radians per minute}$

2. Right triangle setup: The distance from the lighthouse to point $P$ is 600 m (vertical distance). The distance from $P$ to $Q$ is 300 m.

   At any given time, the position of the light on the shore, measured from $P$, is denoted as $x(t)$. The distance from the lighthouse to the point where the beam strikes the shore is: $r = \sqrt{600^2 + x^2}$

   Using trigonometry, the angle $\theta$ between the beam and the shore satisfies: $\tan(\theta) = \frac{x}{600}$ Taking the derivative of both sides with respect to time $t$, we get: $\sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{600} \cdot \frac{dx}{dt}$ But $\frac{d\theta}{dt} = \omega$, so the equation becomes: $\frac{dx}{dt} = 600 \cdot \omega \cdot \sec^2(\theta)$

3. At point $Q$ (where $x = 300$): $\tan(\theta) = \frac{300}{600} = 0.5 \quad \Rightarrow \quad \theta = \tan^{-1}(0.5)$ Using $\sec^2(\theta) = 1 + \tan^2(\theta)$: $\sec^2(\theta) = 1 + 0.5^2 = 1.25$

   Now, the speed of the beam at $Q$ is: $\frac{dx}{dt} = 600 \cdot 6\pi \cdot 1.25$ $\frac{dx}{dt} = 4500\pi \, \text{meters per minute}$ Converting to meters per second (since $1 \, \text{minute} = 60 \, \text{seconds}$): $\frac{dx}{dt} = \frac{4500\pi}{60} \approx 235.62 \, \text{meters per second}$ Rounding to the nearest integer gives $\approx 236 \, \text{meters per second}$.

The speed of the beam along the shore at point $Q$ is approximately 236 meters per second.

Would you like further details on any step, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. What is the beam's speed at the point where $x = 100$ meters?
2. How would the speed change if the lighthouse were further from the shore?
3. How does the speed vary as the beam moves farther away from point $P$?
4. What would happen to the beam's speed if it revolved 4 times per minute instead of 3?
5. How would the problem change if the height of the lighthouse were not neglected?

Tip: When solving related rates problems, always start by finding an equation that connects the variables and differentiate with respect to time.