Find the distance in kilometers between New York and Lima whose latitudes are 41°N and 12°S and the radius of the earth is 6400 kilometers round the answer to the nearest hundred kilometers . Find using arc length formula 

To find the distance between New York and Lima using the arc length formula, we proceed as follows:

The arc length formula is:

\[
d = r \cdot \theta
\]

Where:
- \( d \) is the distance between two points on the surface of the Earth.
- \( r \) is the radius of the Earth.
- \( \theta \) is the central angle between the two points in radians.

### Step 1: Find the central angle \(\theta\)
The latitudes of New York and Lima are:
- New York: \( 41^\circ N \)
- Lima: \( 12^\circ S \)

The total difference in latitude is:

\[
\theta = 41^\circ + 12^\circ = 53^\circ
\]

Since we need the angle in radians, we convert degrees to radians using the conversion factor \( \pi \, \text{radians} = 180^\circ \):

\[
\theta = \frac{53 \times \pi}{180} = 0.924 \, \text{radians}
\]

### Step 2: Apply the arc length formula
Given the Earth's radius \( r = 6400 \, \text{km} \), the distance \( d \) is:

\[
d = 6400 \times 0.924 = 5913.6 \, \text{km}
\]

### Step 3: Round the result
Rounding to the nearest hundred kilometers:

\[
d \approx 5900 \, \text{km}
\]

Thus, the distance between New York and Lima is approximately **5900 kilometers**.

Would you like more details, or do you have any questions? Here are some related questions:

1. How do you convert degrees to radians in other problems?
2. What is the formula to find the central angle between two points using their latitudes?
3. Can this formula be used for points that are not exactly north-south of each other?
4. How would the distance change if the radius of the Earth was larger?
5. What role does Earth's curvature play in calculating large distances?

**Tip**: Always remember that converting angles from degrees to radians is essential for using trigonometric formulas in real-world problems!

Find the distance in kilometers between New York and Lima whose latitudes are 41°N and 12°S and the radius of the Earth is 6400 kilometers. Round the answer to the nearest hundred kilometers. Find using the arc length formula.

Discover how to calculate the distance between New York (41°N) and Lima (12°S) using the arc length formula. This problem explores key concepts in geometry and trigonometry, including converting degrees to radians and applying the formula for arc length. The Earth's radius is 6400 kilometers, and the solution is rounded to the nearest hundred kilometers. Suitable for high school students in Grades 9-12.

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