Problem 71 - Precalculus Mathematics for Calculus

Question

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 mi.) (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The distance between Pittsburgh and Miami is approximately 1036.17 miles.

Step 1: Understand the Problem

We are given that Pittsburgh, PA, and Miami, FL, lie on the same meridian. We are to find the distance between them using their latitudes and the Earth's radius.

Step 2: Determine the Angular Difference

Since both cities lie on the same meridian, only the latitude change needs to be considered. Calculate the angular difference: \[\Delta \phi = 40.5^{\circ} - 25.5^{\circ} = 15^{\circ}\]

Step 3: Convert Angular Difference to Radians

The formula to convert degrees to radians is \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\] Thus, \[15^{\circ} \times \frac{\pi}{180} = \frac{15\pi}{180} = \frac{\pi}{12}\] radians.

Step 4: Calculate the Distance Using Arc Length Formula

The distance between the two cities can be found using the arc length formula: \[\text{Distance} = \theta \times r\] where \( \theta \) is in radians and \( r = 3960 \) miles. So, \[\text{Distance} = \frac{\pi}{12} \times 3960\] Simplifying, \[\text{Distance} = 330\pi \approx 1036.17 \text{ miles}\]

Step 5: Verify the Calculation

Re-check each calculation to ensure no mistakes were made. Confirm that the conversion to radians and distance formula were applied correctly. The result seems accurate based on the calculations completed.

Key Concepts

Angular Difference

When we want to figure out the distance between two places on Earth that lie along the same meridian, identifying their angular difference in latitude is our first important step. The angular difference measures how far apart these two locations are in terms of latitude, and is expressed in degrees.

In our exercise about Pittsburgh and Miami, the two cities lie on nearly the same north-south line along the globe. Pittsburgh’s latitude is given as 40.5°N while Miami’s is 25.5°N. To find the angular difference, simply subtract the two latitudes:

• Angular Difference = Latitude of Pittsburgh - Latitude of Miami
• Angular Difference = 40.5° - 25.5°

This results in an angular difference of 15° which denotes how many degrees of latitude separate the two cities. Keeping these angles in mind helps us move to the next critical step toward finding their geographical distance.

Radians Conversion

When dealing with circular directions, converting measures from degrees to radians is essential. This is because radians are a more natural measure for angles in many mathematical calculations, and particularly when involving circles like Earth's large longitude lines.

To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This allows us to translate the degree measurements into radian measures efficiently. In our example:

• Convert 15° to radians: \( 15^{\circ} \times \frac{\pi}{180} = \frac{\pi}{12} \)

This conversion is crucial when we apply formulas that use radians, ensuring the mathematical equations work properly. This step centers the angle's measure into a form that's easy to use in further geographic calculations.

Arc Length Formula

The arc length formula bridges the concept of geometry with practical applications. When you have an angle in radians and the radius of a circle, you can find the length of the arc subtended by the angle using this formula. It is expressed as:

• Arc Length = \( \theta \times r \)

where \( \theta \) is the angular difference in radians and \( r \) is the Earth's radius given in the problem (3960 miles in this case).

Applying the arc length formula in this exercise:

• \( \text{Distance} = \frac{\pi}{12} \times 3960 \)
• \( \text{Distance} = 330\pi \)

Approximately, this translates to 1036.17 miles. The arc length formula doesn't just compute theoretical outcomes; it stakes a real-world application by defining the curve of Earth's surface between the two cities.

Geographical Distance Calculation

Calculating geographical distance involves blending spherical geometry with real world scenarios. The exercise of using latitude differences to find distance on Earth relies on understanding Earth's shape as a sphere and the need to calculate over its curved surface.

To get the geographical distance, one must:

• Identify the difference in latitude between two locations.
• Convert this angular difference from degrees to radians.
• Apply the arc length formula, considering the Earth's radius for accurate results.

The given scenario simplifies many real-world complexities by assuming Earth's surface along the meridian is a perfect circle. However, this approximation is quite reliable for simple long-distance calculations like between Pittsburgh and Miami. Such calculations form a basis for more complex geographical and navigational solutions, providing foundational insight into earth science and navigation.