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 between Dubuque, Iowa, and Guatemala City is approximately \(4983.129\) km.

Step-by-step solution

Identify given information

In this problem, we are given the latitude coordinates of two cities, and they are on the same longitude. The Earth is assumed to be a sphere. Dubuque, Iowa: Latitude = \(42.50^\circ\) Guatemala City: Latitude = \(14.62^\circ\)

Convert degrees to radians

To calculate the great-circle distance between the two cities, we need to convert the given latitude values from degrees to radians. Radians = Degrees * \(\frac{\pi}{180}\) Dubuque, Iowa (radians): Latitude = \(42.50^\circ * \frac{\pi}{180} = 0.741765\) Guatemala City (radians): Latitude = \(14.62^\circ * \frac{\pi}{180} = 0.255016\)

Great-circle distance formula

Now, we use the great-circle distance formula to find the distance between the two points on Earth's surface. Great-circle distance = \(R * \arccos(\sin(\text{latitude}_1) * \sin(\text{latitude}_2) + \cos(\text{latitude}_1) * \cos(\text{latitude}_2) * \cos(\Delta\text{longitude}))\) Since the cities are on the same longitude, the difference in longitude is \(0^\circ\) which becomes \(0\) radians. Note that the Earth's mean radius, \(R\), is approximately \(6,371\) km.

Calculate the distance

Calculate the distance using the formula and the values we found. Distance = \(6371 * \arccos(\sin(0.741765) * \sin(0.255016) + \cos(0.741765) * \cos(0.255016) * \cos(0))\) Distance = \(6371 * \arccos(0.677498)\) Distance = \(6371 * 0.782183\) Distance = \(4983.129\) km So, the distance between Dubuque, Iowa, and Guatemala City, measured on the Earth's surface, is approximately \(4983.129\) km.

Key concepts

Latitude and Longitude

Understanding the globe means starting with the basics of latitude and longitude. These terms are like the Earth's navigational coordinates and are measured in degrees. Latitude lines are imaginary lines that run parallel to the Equator, dividing the Earth into the Northern and Southern Hemispheres. They range from 0° at the Equator to 90° at the poles.

Longitude lines, or meridians, run from the North Pole to the South Pole and are perpendicular to the Equator. The prime meridian, which is set at 0° longitude, runs through Greenwich, England. Together, the latitude and longitude of any place on Earth can be used to pinpoint an exact location.

For example, a location with latitude 42.50°N and longitude 90° W (like Dubuque, Iowa) is north of the Equator and west of the prime meridian. These coordinates are crucial for calculating distances on the spherical Earth model.

Spherical Earth Model

Our planet is not a perfect sphere, but for many calculations, we use a simplified model and assume it is. This spherical Earth model allows us to use geometry to estimate distances across its surface accurately, even though it has some irregularities like mountains and valleys.

When calculating distances, we refer to 'great-circle distances,' which are the shortest paths between two points on the globe. Imagine slicing through the Earth's center; the edge of that slice is a great circle. This concept becomes crucial when determining the most efficient routes between two points on the Earth's surface, such as in aviation and maritime navigation.

Moreover, this model helps us translate a three-dimensional problem into a two-dimensional one, simplifying calculations without significant losses in accuracy for most practical purposes.

Radian Conversion

Radians are a unit of angular measure used in many areas of mathematics, including the calculation of great-circle distances. One radian is the angle made at the center of a circle by an arc whose length is equal to the circle's radius. In comparison to degrees, there are about 6.28318 (2π) radians in a full circle (360°).

In our calculations, we need to convert degrees to radians to use the trigonometric functions properly, because these functions inherently use radians in their computation. The conversion is straightforward: to convert degrees to radians, multiply by π/180.

For example, a latitude of 42.50° is converted to radians by multiplying 42.50° by π/180, resulting in approximately 0.741765 radians. This step is critical to ensure accurate calculations when applying trigonometry.

Earth's Mean Radius

Knowing the Earth's size is essential when calculating distances between two points on its surface. For many calculations, we use the Earth's mean radius, which is the average radius considering the Earth's spheroid shape. The Earth's mean radius is about 6,371 kilometers (3,959 miles).

The use of the mean radius simplifies the calculations needed for determining distances, yet it is accurate enough for most applications. In the context of calculating great-circle distances, the mean radius becomes a scaling factor that helps convert angular distances (in radians) into linear distances (in kilometers or miles).

The formula takes the central angle in radians between our two points and multiplies it by the Earth's mean radius to calculate the distance across the Earth's surface. Without the Earth's mean radius, we would not be able to translate the angular measurements into tangible distances.