Let's break down the problem.

Problem Analysis

A lighthouse stands $475 \, \text{m}$ from a straight shore, with its light beam revolving at a constant rate of two rotations per minute. We have:

• The point $P$ is on the shore closest to the lighthouse.
• The point $Q$ is $225 \, \text{m}$ from $P$ along the shore.
• We are asked to find the speed of the beam along the shore when it strikes point $Q$.

Approach

1. Angular Speed: Since the beam completes two full rotations per minute, its angular speed $\omega$ is: $\omega = 2 \times 2\pi \, \text{radians/min} = 4\pi \, \text{radians/min}$

2. Using Trigonometry for Distance: Let $\theta$ be the angle between the line from the lighthouse to $P$ and the line from the lighthouse to $Q$. Then, $\tan(\theta) = \frac{225}{475}$ Solving for $\theta$ will give us the position where the beam hits $Q$.

3. Differentiating to Find Speed Along the Shore: The distance along the shore where the beam hits can be represented as $x = 475 \tan(\theta)$. Taking the derivative with respect to time $t$, we get: $\frac{dx}{dt} = 475 \sec^2(\theta) \cdot \frac{d\theta}{dt}$ Since $\frac{d\theta}{dt} = \omega = 4\pi$, substitute this to find $\frac{dx}{dt}$ at the point $Q$.

4. Final Calculation: Substitute the values to calculate the speed along the shore, and then round to the nearest integer.

I’ll work through the calculations now.The speed of the beam along the shore when it strikes point $Q$ is approximately 7308 meters per minute.

Would you like a more detailed breakdown of any step or have further questions?

Here are some related questions to explore:

1. How does changing the distance of the lighthouse from the shore affect the beam speed along the shore?
2. What happens to the speed along the shore if the angular speed of the light increases?
3. How would the speed change if point $Q$ were located farther from point $P$?
4. Can this problem be solved if the light's rotational speed varied instead of being constant?
5. What would be the effect on speed if the lighthouse were at a different height?

Tip: Remember, the relationship between angular and linear speed depends on the distance from the rotation center, making the linear speed grow as distance increases.