Find the distance along an arc on the surface of the earth that subtends a central angle of 3 minutes (1 minute = 1/60 degree). The radius of the earth is 3960 miles. Round to the thousandths. (3 decimal places)

To find the distance along an arc on the surface of the Earth that subtends a central angle, we can use the following formula:

\[
\text{Arc length} = r \theta
\]

where:
- \( r \) is the radius of the Earth,
- \( \theta \) is the central angle in radians.

### Step 1: Convert the central angle from minutes to radians.

First, we need to convert the angle from minutes to degrees:
\[
3 \text{ minutes} = \frac{3}{60} \text{ degrees} = 0.05 \text{ degrees}
\]

Next, convert the angle from degrees to radians using the conversion factor \( \frac{\pi}{180} \):

\[
\theta = 0.05 \times \frac{\pi}{180} \text{ radians} = \frac{0.05\pi}{180} \text{ radians}
\]

### Step 2: Calculate the arc length.

Now, we can substitute the values into the formula:
\[
\text{Arc length} = 3960 \times \frac{0.05\pi}{180} 
\]

### Step 3: Compute the result.

Let's calculate the arc length:

\[
\text{Arc length} \approx 3960 \times \frac{0.05\pi}{180} \approx 3960 \times 0.00087266 \approx 3.456 \text{ miles}
\]

### Final Answer:

The distance along the arc is approximately **3.456 miles**.

Would you like more details on any of these steps or have any other questions?

Here are five related questions you might find interesting:
1. How would the arc length change if the central angle was 6 minutes instead of 3?
2. What is the circumference of the Earth using the given radius?
3. How would you find the arc length if the radius of the Earth were given in kilometers?
4. How does converting angles from degrees to radians help in solving arc length problems?
5. How would you calculate the area of the sector formed by a central angle of 3 minutes?

**Tip:** Remember, when dealing with arc length problems, always ensure the angle is in radians to apply the formula \( s = r \theta \) correctly!

Learn how to calculate the distance along an arc on the surface of the Earth that subtends a central angle of 3 minutes. Use the formula s = r * theta where the radius of the Earth is 3960 miles. Explore step-by-step solution and related questions.

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