Problem 72 - Precalculus Mathematics for Calculus

Question

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 mi.)

Step-by-Step Solution

Answer

The distance between Memphis and New Orleans is approximately 345.66 miles.

Step 1: Understand the Problem

We need to find the distance between two cities located on the same meridian using their latitudes. Memphis is at a latitude of \(35^{\circ} \, \mathrm{N}\) and New Orleans at \(30^{\circ} \, \mathrm{N}\). The radius of the Earth is given as 3960 miles.

Step 2: Calculate the Angular Difference

Calculate the difference in latitude since the cities are on the same meridian. This gives us the angular separation in degrees: \(35^{\circ} - 30^{\circ} = 5^{\circ}.\)

Step 3: Convert Angular Difference to Radians

Use the conversion factor \(\pi\) radians = \(180^{\circ}\) and find the angular difference in radians: \(5^{\circ} \times \frac{\pi}{180}\). This results in \(\frac{\pi}{36}\) radians.

Step 4: Calculate Arc Length

The distance between the two cities is given by the arc length formula: \(\text{Arc Length} = \text{Radius} \times \text{Angle in radians}\). Substituting the known values, we obtain: \(3960 \times \frac{\pi}{36}\).

Step 5: Perform the Calculation

Compute the arc length: \(3960 \times \frac{\pi}{36} \approx 3960 \times 0.0873 = 345.66\) miles.

Key Concepts

Meridian Distance Calculation

To find the distance between two points on the Earth that lie along the same meridian, we calculate the meridian distance. This calculation is essentially determining the arc length along the Earth's surface between two points with different latitudes.

In our scenario, both Memphis and New Orleans are on the same meridian, which simplifies the process since we only need to look at their latitudinal positions. The first step is to find the difference in these latitudes. This difference will serve as the angular separation we'll later convert to radians.

Using the formula for arc length ensures we account for the Earth's curvature. This is important because the Earth is not a flat surface, and the shortest path along the ground is not a straightforward line, but an arc. By focusing on the Earth's radius and angular separation, we leverage the properties of circles to find precise distances on the Earth's surface.

Arc Length Formula in Radians

The arc length formula is crucial in calculating distances along the Earth's surface. This formula takes advantage of the fact that Earth can be modeled as a sphere. The formula is:

• Arc Length = Radius of Earth × Angle in Radians

To find the angle in radians, we first need to convert the angular separation from degrees to radians. This is because radians are a more natural measure when dealing with arc lengths.

We use the conversion factor:

• \(1^{\circ} = \frac{\pi}{180}\ text{ radians}\)

This allows us to easily switch from degrees, which are often more intuitive, to radians, which are more suitable for calculations in circular geometry. Once we have the angle in radians, calculating the arc length becomes a straightforward multiplication.

Latitude Angular Separation

Latitude angular separation is the key measure we need to calculate distances between two points on the same meridian. It refers to the difference in their latitude degrees, which need to be converted into radians.

In our example, the latitudinal separation (35° – 30°) equates to an angular separation of 5°. This small difference, while seemingly insignificant in degrees, corresponds to a specific arc length when using the Earth's radius for calculation. So why bother?

• Degrees are better for graphical displays and simple communication.
• Radians are better for precise mathematical calculations like arc lengths.

This conversion and calculation result in the actual distance one might travel if moving along the same meridian, which is an arc on the Earth's surface. Understanding this concept allows the easy translation of geographical data into usable engineering and navigational information.