A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes seven revolutions per minute. How fast (in km/min) is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)


To solve this problem, we need to use related rates. The setup involves a rotating beam of light sweeping across a shoreline. We can approach it geometrically by understanding that the light beam's position relative to the shoreline changes as it rotates, and we're asked to determine how fast the beam is moving along the shoreline when the light is 1 km from point P.

### Step-by-Step Solution

1. **Visualize the problem:**
   - The lighthouse is 4 km away from point P (the closest point on the shoreline).
   - The light rotates at 7 revolutions per minute.

2. **Define the variables:**
   - Let \( \theta \) be the angle between the line from the lighthouse to the point P and the beam of light at some time \( t \).
   - Let \( x \) be the distance of the beam's position along the shoreline from point P at time \( t \).

   By trigonometry, we know:
   \[
   \tan(\theta) = \frac{x}{4}
   \]

3. **Differentiate with respect to time:**
   We need to find how fast \( x \) is changing with respect to time. So, differentiate both sides of the equation \( \tan(\theta) = \frac{x}{4} \) with respect to time \( t \):
   \[
   \sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{4} \cdot \frac{dx}{dt}
   \]
   Here, \( \frac{dx}{dt} \) is the speed of the beam along the shoreline that we are looking for, and \( \frac{d\theta}{dt} \) is the angular speed of the light.

4. **Angular speed:**
   The lighthouse makes 7 revolutions per minute, so its angular speed is:
   \[
   \frac{d\theta}{dt} = 7 \times 2\pi = 14\pi \text{ radians per minute}
   \]

5. **Evaluate when the light is 1 km away:**
   When the beam of light is 1 km away from point P, \( x = 1 \). At this moment, we can find \( \theta \) using:
   \[
   \tan(\theta) = \frac{1}{4} \quad \Rightarrow \quad \theta = \tan^{-1}\left(\frac{1}{4}\right)
   \]
   Now, find \( \sec^2(\theta) \). Recall that:
   \[
   \sec^2(\theta) = 1 + \tan^2(\theta)
   \]
   Substituting \( \tan(\theta) = \frac{1}{4} \):
   \[
   \sec^2(\theta) = 1 + \left(\frac{1}{4}\right)^2 = 1 + \frac{1}{16} = \frac{17}{16}
   \]

6. **Solve for \( \frac{dx}{dt} \):**
   Now substitute everything into the differentiated equation:
   \[
   \frac{17}{16} \cdot 14\pi = \frac{1}{4} \cdot \frac{dx}{dt}
   \]
   Solving for \( \frac{dx}{dt} \):
   \[
   \frac{dx}{dt} = \frac{17 \cdot 14\pi}{4} \times \frac{16}{17} = 14\pi \times 4
   \]
   \[
   \frac{dx}{dt} = 56\pi \, \text{km/min}
   \]

7. **Final answer:**
   Now approximate \( \pi \approx 3.1416 \):
   \[
   \frac{dx}{dt} \approx 56 \times 3.1416 = 175.9 \, \text{km/min}
   \]

So, the beam of light is moving along the shoreline at approximately **175.9 km/min** when it is 1 km away from point P.

Let me know if you want any details or have further questions!

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Here are 5 related questions:
1. What is the speed of the light beam along the shoreline when it is 2 km from point P?
2. How would the answer change if the lighthouse were 5 km away from point P instead of 4 km?
3. What happens to the speed of the beam as the distance from P increases?
4. Can we calculate the angular speed in degrees per minute instead of radians per minute?
5. How does the problem change if the rotation speed of the lighthouse increases to 10 revolutions per minute?

**Tip:** The further the beam is from the lighthouse, the faster it moves along the shoreline due to the nature of angular motion and its relationship to linear velocity!

A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes seven revolutions per minute. How fast (in km/min) is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)

Learn how to calculate the speed of a lighthouse beam along a shoreline using related rates and trigonometry. The light makes seven revolutions per minute and is 1 km away from the nearest point on the shore. This detailed step-by-step solution explores key concepts like angular speed and trigonometric identities. Suitable for advanced high school or college-level calculus students.

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