Problem 70 - Algebra and Trigonometry

Question

Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has latitude \(35^{\circ} \mathrm{N}\) and New Orleans, \(30^{\circ} \mathrm{N}\) . Find the distance between these two cities. (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Step-by-Step Solution

Answer

The distance is approximately 345 miles.

Step 1: Understand the Earth's Circle

The Earth is approximately a sphere, so we use the formula for the arc length of a circle section. Given that the latitudes represent angles on this circle, Memphis and New Orleans lie along the same meridian.

Step 2: Calculate Latitude Difference

Find the difference in latitude between Memphis and New Orleans. Both are given in degrees: \[ 35^{\circ} - 30^{\circ} = 5^{\circ} \]So, the difference in latitude is \(5^{\circ}\).

Step 3: Convert Degrees to Radians

To calculate the arc length or distance, convert the angle from degrees to radians, because the formula requires radians. The conversion factor is \(\pi/180^{\circ}\): \[ \text{Latitude in radians} = 5^{\circ} \times \frac{\pi}{180} \approx 0.0873 \, \text{radians} \]

Step 4: Apply Arc Length Formula

The formula for the arc length \( s \) of a circle is \( s = r \theta \), where \( r \) is the radius of the Earth (3960 mi), and \( \theta \) is the angle in radians:\[ s = 3960 \times 0.0873 \approx 345.348 \, \text{miles} \]

Step 5: Interpret the Result

The calculated arc length represents the distance between Memphis and New Orleans along the Earth's surface, given the radius and latitude difference.

Key Concepts

Arc Length

The concept of arc length is central to understanding distances along curves or circular paths. When dealing with circular objects like the Earth, the arc length refers to the curved distance along the surface.
In trigonometry, to find the arc length, we use the formula:

• \( s = r \theta \)

Where:

• \( s \) is the arc length.
• \( r \) is the radius of the circle (or sphere, in this context).
• \( \theta \) is the angle in radians.

This formula is particularly useful for calculating distances between two points on the same circular path, such as cities on the Earth that lie along the same meridian. The arc length considers the curvature of the Earth, providing a more accurate measure compared to straight-line distances.

Latitude

Latitude measures how far north or south a location is from the equator. It's expressed in degrees, where the equator is 0 degrees, and the poles are at 90 degrees. Each degree of latitude represents approximately 69 miles.
In the context of this exercise, Memphis at 35°N and New Orleans at 30°N have a latitude difference of 5°. This difference is the angle of the circular path along the Earth's surface that connects these two cities.
Latitude is crucial for navigation and geography, helping to determine how far apart locations are, especially those on the same meridian or similar longitudinal lines.

Radians

Radians are another way to measure angles, and they are critical in calculating arc lengths and other trigonometric functions. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of the circle.
One full circle is equal to \( 2\pi \) radians. Therefore, to convert degrees to radians, you use the formula:

• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)

In our exercise, the latitude difference of 5° was converted to radians for use in the arc length formula. This conversion worked out to approximately 0.0873 radians. Using radians simplifies complex calculations involving circles, as it relates directly to the radius.

Earth's Radius

The Earth's radius is a key element when dealing with calculations involving latitude and arc length. The average radius of the Earth is about 3960 miles. This measurement represents the distance from the Earth's center to its surface and is crucial for determining distances over the globe.
In problems involving arc length, like the exercise, the Earth's radius is used in the formula \( s = r\theta \).

• The radius \( r \) helps to comprehend the scaling of the Earth's curvature when calculating the distance along a meridian.
• The value is essential for obtaining accurate results, which reflect real-world distances between locations.

Using the Earth's radius ensures that we account for its spherical shape rather than treating it as a flat surface.