Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Find the distance in kilometers. between the pair of cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of Earth is 6400 kilometers. Round answers to the nearest hundred kilometers. Charleston, South Carolina, \(33^{\circ} \mathrm{N},\) and Toronto, Ontario, \(43^{\circ} \mathrm{N}\)

Step-by-Step Solution

Answer

The distance is 1100 kilometers.

Step 1: Determine Latitude Difference

Identify the latitudinal coordinates of both cities. Charleston is at \(33^{\circ} \mathrm{N}\) and Toronto is at \(43^{\circ} \mathrm{N}\). Calculate the difference in latitude: \(43^{\circ} - 33^{\circ} = 10^{\circ}\).

Step 2: Convert Latitude Difference to Radians

Convert the difference in degrees to radians for distance calculation. Use the conversion: \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\). Thus, \(10^{\circ} \times \frac{\pi}{180} = \frac{\pi}{18}\) radians.

Step 3: Calculate Arc Distance Based on Earth's Radius

The formula to calculate the distance on a spherical surface is \(D = R \times \theta\), where \(D\) is the distance, \(R\) is the radius of the Earth, and \(\theta\) is the angle in radians. Substitute \(R = 6400\) km and \(\theta = \frac{\pi}{18}\):\[D = 6400 \times \frac{\pi}{18}\approx 1117.06 \text{ km}\].

Step 4: Round the Distance

Round the distance calculated in Step 3 to the nearest hundred kilometers. Therefore, \(1117.06 \text{ km}\) rounds to \(1100 \text{ km}\).

Key Concepts

Latitude and Longitude

Latitude and longitude are crucial for pinpointing locations on Earth. Both of these elements form the geographic coordinate system used worldwide. Latitude measures how far north or south a location is from the Equator, and is indicated in degrees north (N) or south (S). Longitude, on the other hand, measures how far east or west a location is from the Prime Meridian. For example, Charleston is at \(33^{\circ} \mathrm{N}\), meaning it's 33 degrees north of the Equator. In contrast, Toronto is at \(43^{\circ} \mathrm{N}\), placing it 43 degrees north.
Knowing the latitude of two points allows us to calculate the difference between them, which is vital for distance calculations, especially if they lie on the same north-south line, as is the case in this exercise.

Radian Conversion

Radian conversion is a critical step when calculating distances on Earth. Unlike degrees, radians measure angles in terms of the radius of a circle. To calculate the distance between two points using spherical geometry, angles must first be expressed in radians.

• To convert degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).
• This formula stems from the relationship that \(180^{\circ}\) is equivalent to \(\pi\) radians.

In our exercise, the latitude difference of \(10^{\circ}\) becomes \(\frac{\pi}{18}\) after conversion. Always remember to use radians in formulas involving distances on curved surfaces to ensure accurate results.

Spherical Geometry

Spherical geometry is used when dealing with shapes on the surface of a sphere, like Earth. This type of geometry differs from plain Euclidean geometry, where the assumptions and rules are somewhat different due to Earth's curvature.
The key equation for distance on Earth's surface is derived from the arc length formula: \(D = R \times \theta\), where:

• \(D\) is the distance.
• \(R\) is the radius of the sphere (Earth, in this case).
• \(\theta\) is the central angle in radians.

In our specific problem, given Earth's radius \(6400\) km and an angle of \(\frac{\pi}{18}\), this formula helps us calculate the direct path along Earth's surface, a crucial measurement in navigation and geography.

Earth's Radius

Earth's radius is central to understanding distance calculations on its surface. Essentially, it is the distance from Earth's center to any point on its surface, averaging around 6,400 kilometers. This mean radius is pivotal when applying the spherical geometry formula \(D = R \times \theta\).
Using Earth's radius, we can compute the arc distance between longitudinal or latitudinal positions. When performing such calculations, assume a simplified model where Earth is a perfect sphere.

• This approximation makes our math easier while retaining practical accuracy.
• It's important to note Earth's radius slightly varies depending on location, but \(6,400\) km serves as a useful mean value.

Calculating distances minus this fundamental value would be near impossible, exemplifying its importance in geographic computations.