Problem 71 - Algebra and Trigonometry

Question

Latitudes Pittsburgh, Pennsylvania, and Miami, Florida, lie approximately on the same meridian. Pittsburgh has a latitude of \(40.5^{\circ} \mathrm{N},\) and Miami has a latitude of \(25.5^{\circ} \mathrm{N} .\) Find the distance between these two cities. (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Step-by-Step Solution

Answer

The distance between Pittsburgh and Miami is approximately 1035.6 miles.

Step 1: Determine the Difference in Latitudes

Subtract Miami's latitude from Pittsburgh's latitude to find the angular distance between the two cities. This is given by \( 40.5^\circ - 25.5^\circ = 15^\circ \).

Step 2: Convert Angular Distance to Radians

To calculate the distance along the Earth's surface, convert the angular difference from degrees to radians using the conversion formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This gives us \( 15^\circ \times \frac{\pi}{180} = \frac{\pi}{12} \) radians.

Step 3: Calculate the Arc Length

The distance between the two cities along the Earth's surface (arc length) is calculated by multiplying the Earth's radius by the angular distance in radians: \( \text{distance} = \text{radius} \times \text{angular distance} \). Substituting the values, we have \( 3960 \, \text{mi} \times \frac{\pi}{12} \approx 1035.6 \, \text{mi} \).

Key Concepts

Latitude

Latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. This concept is important because it helps us understand how locations are positioned relative to each other on the globe. Latitude is measured in degrees ( https://en.wikipedia.org/wiki/Degree_(angle)). These degrees range from 0° at the Equator to 90° at the poles.

Here are a few key points worth noting about latitude:

• Equator's Special Position: The equator is at 0° latitude, dividing the Earth into the Northern and Southern Hemispheres.
• Latitude Lines: Lines of latitude are parallel and are sometimes referred to as "parallels." They remain equidistant from each other.
• Latitude vs. Longitude: Do not confuse latitude with longitude, which runs from pole to pole and measures the east-west positioning.
• Impact on Climate: Latitude influences climate as areas near the equator are typically warmer, while areas near the poles are colder.

In our exercise, Pittsburgh and Miami's latitudes determine their positions on this geographic coordinate system, allowing us to calculate how far apart they are.

Angular Distance

Angular distance is the measure of the "angle" between two points on the Earth's surface as seen from the Earth's center. This concept is crucial when calculating distances on a sphere like Earth. It is typically expressed in degrees or radians.

Calculating Angular Distance

• Determine Differences: First, find latitude differences between two points. In our example, Pittsburgh is at 40.5°N and Miami at 25.5°N. The difference is 15°.
• Radian Conversion: Degrees alone might not be handy for further math calculations like finding actual distances, so we convert these degrees into radians using the formula: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}.\]
• Conversion Example: With our 15° difference, we have:\[15 \times \frac{\pi}{180} = \frac{\pi}{12} \text{ radians.}\]

This radian measure allows us to understand how large the arc is on Earth's sphere, facilitating the calculation of actual surface distances.

Arc Length Calculation

Arc length calculation transforms angular distance into a measurable linear distance on the spherical surface. This is particularly vital in geographical and navigational contexts.

To Calculate the Arc Length:

• Core Formula: The formula for arc length (distance) is given as:\[\text{Arc Length} = \text{radius} \times \text{angular distance (in radians)},\]where radius refers to Earth's radius (3960 miles in our exercise).
• Application: Use the formula with the radius (3960 mi) and angular distance (\[\frac{\pi}{12}\]), calculated previously:\[\text{Arc Length} = 3960 \times \frac{\pi}{12} \approx 1035.6 \text{ miles.}\]
• Understanding the Result: The result represents the physical distance along the Earth's surface between Pittsburgh and Miami.

This straightforward calculation empowers us to convert an angular measurement into an easily understandable linear distance, making practical application of spherical geometry possible.