Problem 72 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The distance between Memphis and New Orleans is approximately 346 miles.

Step 1: Identify the Change in Latitude

The change in latitude between Memphis and New Orleans is calculated by subtracting the latitude of New Orleans from the latitude of Memphis. Since both cities are on the same meridian, the change in latitude is simply: \(35^{\circ} - 30^{\circ} = 5^{\circ}\).

Step 2: Convert Degrees to Radians

To use the distance formula that involves the radius, convert the latitude change from degrees to radians. The conversion formula is: \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\). Therefore, \(5^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{180} = \frac{\pi}{36}\) radians.

Step 3: Calculate the Arc Length

Use the formula for arc length, \( S = r \theta \), where \(r\) is the Earth's radius (3960 miles) and \(\theta\) is the angle in radians. Plug the values into the equation: \( S = 3960 \times \frac{\pi}{36} \).

Step 4: Simplify to Find the Distance

Perform the multiplication to find the arc length, which is the distance between the two cities: \( S = 3960 \times \frac{\pi}{36} \approx 3960 \times 0.0873 \approx 345.6 \). Thus, the distance is approximately 346 miles.

Key Concepts

Degrees to Radians Conversion

When calculating distances on the Earth's surface, particularly when navigating between two latitudes, converting degrees to radians is an essential step. This is because most trigonometric and calculus-based distance formulas, like the arc length formula, require angular measurements in radians. So, what exactly is a radian?
A radian measures angles based on the radius of a circle. There are approximately 6.283 radians, or roughly 2π radians, in a full circle, as opposed to the 360 degrees we've grown accustomed to in daily life. To convert degrees to radians, use the formula:

• \[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]

This formula multiplies the degree measure by the factor \( \frac{\pi}{180} \), effectively converting 180 degrees to \( \pi \) radians. For example, to convert the change in latitude between Memphis and New Orleans (5 degrees) to radians, you multiply 5 by \( \frac{\pi}{180} \), resulting in \( \frac{\pi}{36} \) radians. This conversion is crucial for precise calculations.

Arc Length Formula

The arc length formula is a fundamental concept when calculating distances along the surface of the Earth, which is considered circular for simplicity in this context. The formula is represented mathematically as:

• \[ S = r \theta \]

Here, \( S \) is the arc length, \( r \) is the radius of the circle (or the Earth's radius, in our case), and \( \theta \) is the angle in radians subtended by the arc at the center of the circle.
When you use this formula, imagine slicing a piece of pizza. The crust length of that slice is akin to our arc length, while the pizza's radius corresponds to the Earth's radius. Now, when calculating the distance between Memphis and New Orleans, the Earth's radius is 3960 miles, and the angle, after converting to radians, is \( \frac{\pi}{36} \). Substituting these values into the arc length formula gives:

• \[ S = 3960 \times \frac{\pi}{36} \]

This calculates an arc length of approximately 346 miles, which is the distance between the two cities. Understanding and using the arc length formula allows one to grasp how geometrical properties translate into navigational insights.

Latitude and Longitude

Latitude and longitude are vital components of the geographic coordinate system. This system is used to specify every location on Earth with a set of numbers. Latitude refers to the north-south positioning on the Earth's surface, while longitude refers to the east-west positioning.

• **Latitude** is measured in degrees from the Equator, ranging from 0° at the Equator to 90° at the poles (north or south).
• **Longitude** is measured in degrees from the Prime Meridian, which is the 0° measurement line running through Greenwich, England, to 180° east or west.

In the context of distance calculation discussed here, both Memphis, Tennessee, and New Orleans, Louisiana, share approximately the same line of longitude. This means they lie along the same meridian, simplifying calculations as we only focus on changes in latitude to estimate distances.
With a latitude of 35°N for Memphis, and 30°N for New Orleans, the calculation focuses on the difference in latitude, disregarding longitude changes. Such measurements enable precise navigation and distance estimation across our planet's surface, essential for travel, geography, and mapping.