Question

Assuming that the Earth is spherical and recalling that latitudes range from \(0^{\circ}\) at the Equator to \(90^{\circ} \mathrm{N}\) at the North Pole, how far apart, measured on the Earth's surface, are Dubuque, Iowa \(\left(42.50^{\circ} \mathrm{N}\right.\) latitude \()\), and Guatemala City \(\left(14.62^{\circ} \mathrm{N}\right.\) latitude \() ?\) The two cities lie on approximately the same longitude. Do not neglect the curvature of the Earth in determining this distance.

Short answer

Answer: The distance is approximately 3100.81 km.

Step-by-step solution

Find the difference in latitudes

First, we need to find the difference in latitudes between the two cities. Subtract the latitude of Guatemala City from the latitude of Dubuque, Iowa. \(42.50^{\circ} - 14.62^{\circ} = 27.88^{\circ}\) So, the difference in latitudes is \(27.88^{\circ}\).

Convert degrees to radians

To use the arc length formula, we need to convert the difference in latitudes from degrees to radians. The conversion formula is: Radians = Degrees × \(\frac{\pi}{180}\) So, we have: \(27.88^{\circ} \times \frac{\pi}{180} = 0.4867\) radians

Use the arc length formula

Now we need to use the arc length formula on a sphere to find the distance between the two cities. The formula is: Distance = Radius × Angle The average radius of the Earth is approximately 6371 km. Using the angle (in radians) we've just calculated, we can find the distance: Distance = \(6371\,\text{km} \times 0.4867\,\text{radians}\)

Calculate the distance

Now, we can calculate the distance between Dubuque, Iowa and Guatemala City: Distance = \(6371\,\text{km} \times 0.4867\,\text{radians} = 3100.81\,\text{km}\) So, the distance between the two cities, measured on the Earth's surface, is approximately 3100.81 km.

Key concepts

Latitude and Longitude

Latitude and longitude are part of a geographic coordinate system that is used to pinpoint locations on the Earth's surface. Latitude lines, also called parallels, circle the earth parallel to the equator and measure how far north or south a location is from this central line. They are measured in degrees, from the Equator at 0° to the poles at 90° North or South.

Longitude lines, or meridians, on the other hand, run perpendicular to latitude lines and measure the distance east or west of the Prime Meridian, which is set at 0° and passes through Greenwich, England. Both latitude and longitude are essential for calculating distances between two points on the Earth, particularly in the context of great circle distances which follow the curvature of the Earth.

Converting Degrees to Radians

To calculate great circle distances, we often need to convert angular measurements from degrees to radians. This is because many mathematical formulas, including the arc length formula, use radians as the unit of measure for angles. The conversion is critical, as degrees and radians are simply different ways of expressing the same thing.

The conversion is straightforward: we multiply the number of degrees by \(\pi / 180\). For instance, an angle of 27.88° would be converted to radians as follows: \(27.88° \times \frac{\pi}{180} = 0.4867 \) radians. By converting degrees to radians, we can jump to the next step of applying the arc length formula effectively.

Arc Length Formula

The arc length formula is the key to determining distances on the globe when we have an angle and the radius of the circle. On the surface of the Earth, the formula looks like this: Distance = Radius × Angle (in radians).

In situations like finding the distance between two cities with the same longitude, we use the difference in their latitudes to represent the angle in the formula. Since the Earth is nearly spherical, these distances represent arcs of great circles, which are the shortest paths between two points on a sphere. Understanding and applying this formula is fundamental in navigation and geography when calculating actual travel distances.

Earth's Curvature

When calculating distances over the Earth's surface, accounting for Earth's curvature is crucial. The Earth is not a perfect sphere but is approximated as one for many calculations. This curvature means that the straight-line distance between two points on a map is not the same as the actual travel distance.

The arc length formula incorporates Earth's curvature by utilizing the Earth's radius—the distance from the center of the Earth to the surface—and the angle between the points, measured in radians. When we refer to the great circle distance, we're talking about the shortest distance between two points along the surface of the sphere, rather than through its interior, which is particularly important for long-distance travel where curvature has a significant effect.