This problem is a classic example of related rates in calculus. We'll break it down step by step.

Step 1: Analyze the problem geometrically

• The lighthouse is 5 km away from point $P$, which is the closest point on the shoreline to the lighthouse.
• The beam of light rotates at 2 revolutions per minute. So, the angular speed $\omega$ is: $\omega = 2 \times 2\pi \, \text{radians/min} = 4\pi \, \text{radians/min}.$

We are asked to find the speed at which the beam of light is moving along the shoreline when the beam is 1 km from point $P$.

Step 2: Set up the relationship

We can use trigonometry to describe the position of the beam along the shoreline. Let:

• $x$ be the distance of the point of light from $P$ along the shoreline.
• $\theta$ be the angle between the line from the lighthouse to $P$ and the current position of the light beam.

Using the geometry of the situation, we have: $\tan(\theta) = \frac{x}{5}.$ Differentiating both sides with respect to time $t$ gives: $\sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{5} \cdot \frac{dx}{dt}.$ We know that $\frac{d\theta}{dt} = \omega = 4\pi$, so the equation becomes: $\sec^2(\theta) \cdot 4\pi = \frac{1}{5} \cdot \frac{dx}{dt}.$ Solving for $\frac{dx}{dt}$, the speed of the beam of light along the shoreline: $\frac{dx}{dt} = 5 \cdot 4\pi \cdot \sec^2(\theta).$

Step 3: Find $\sec^2(\theta)$

When the beam is 1 km from $P$, $x = 1$. From the relationship $\tan(\theta) = \frac{x}{5}$, we have: $\tan(\theta) = \frac{1}{5}.$ Thus, using the identity $\sec^2(\theta) = 1 + \tan^2(\theta)$: $\sec^2(\theta) = 1 + \left(\frac{1}{5}\right)^2 = 1 + \frac{1}{25} = \frac{26}{25}.$

Step 4: Calculate the speed

Now we substitute $\sec^2(\theta) = \frac{26}{25}$ into the expression for $\frac{dx}{dt}$: $\frac{dx}{dt} = 5 \cdot 4\pi \cdot \frac{26}{25} = 4\pi \cdot \frac{26}{5} = \frac{104\pi}{5} \, \text{km/min}.$ Thus, the speed of the light beam along the shoreline when it is 1 km from $P$ is: $\frac{104\pi}{5} \, \text{km/min} \approx 65.34 \, \text{km/min}.$

Final Answer:

The beam of light is moving along the shoreline at approximately $65.34 \, \text{km/min}$ when it is 1 km from point $P$.

Would you like more details or have any questions?

Here are 5 questions to further expand on this topic:

1. What would the speed of the light beam be if it were 2 km from point $P$?
2. How would the speed change if the lighthouse were 10 km from the shoreline instead of 5 km?
3. Can you explain the role of trigonometric identities in related rates problems?
4. How would the speed of the beam change if the light rotated at 3 revolutions per minute instead of 2?
5. What is the significance of the secant function in this problem?

Tip: Always make sure to relate your variables clearly when working with related rates problems—visualizing the geometry can help guide the setup.