Problem 90 - Precalculus : Building Concepts and Connections

Question

Chicago, Illinois (\(42^{\circ}\) north latitude), is due north of Birmingham, Alabama (33" north latitude). If Earth's radius is approximately 3900 miles, find the approximate distance between the two cities, to two decimal places.

Step-by-Step Solution

Answer

The short answer to the problem would come from Step 3's calculations. Given the Earth's radius as 3900 miles and the difference in latitudes as \(9^{\circ}\), the approximate distance between Chicago, Illinois and Birmingham, Alabama is \((9/360) \times 2\pi \times 3900\) miles, calculated to two decimal places.

Step 1: Understand the problem and identify the variables

The problem requires the calculation of the distance between two cities on a sphere (the Earth), knowing their latitudes and the radius of the sphere. We have: Latitude of Chicago, Illinois = \(42^{\circ}\), Latitude of Birmingham, Alabama = \(33^{\circ}\), Radius of the Earth = 3900 miles.

Step 2: Convert the difference in latitudes to a distance

First, we need to find the difference in latitudes between the two cities: \(42^{\circ} - 33^{\circ} = 9^{\circ}\). This difference needs to be converted to a distance. Given that the Earth completes a \(360^{\circ}\) rotation and this corresponds to a distance equal to the Earth’s circumference, we can write the ratio: \(9^{\circ}/360^{\circ}\). To get the distance that corresponds to \(9^{\circ}\), we need to multiply this ratio by the Earth's circumference. We are given that Earth's radius is 3900 miles. The circumference of the Earth at the equator is therefore \(2\pi r = 2\pi \times 3900\) miles.

Step 3: Calculate the distance between the two cities

To find the distance that corresponds to \(9^{\circ}\) of latitude, multiply \(9/360\) times the Earth's circumference, \(2\pi r = 2\pi \times 3900\). Therefore, the distance between the two cities is approximately \((9/360) \times 2\pi \times 3900\) miles. Applying the calculation gives the approximate distance.

Key Concepts

Earth's circumference

The Earth's circumference is a fundamental concept in precalculus, particularly when dealing with geographical positioning and distance calculations.
When we refer to the circumference, we are talking about the full distance around the Earth’s equator, akin to the perimeter of a circle. Given that the Earth is approximately a sphere, this notion allows us to estimate distances as if we are measuring along a circular path.
To calculate the Earth's circumference, you'll need to know its radius. The formula for the circumference of a circle is given by: \ \\[ C = 2\pi r \] \ \Where: \ \

• \(C\) is the circumference,
\
• \(\pi\) (Pi) is a constant approximately equal to 3.14159,
\
• \(r\) is the radius of the Earth.
\

Given the radius of approximately 3900 miles, the Earth's circumference can be calculated as \(2 \pi \times 3900\). Knowing the circumference helps us in understanding how to convert angular distances between points on Earth (like latitude differences) into actual distances.

latitude and longitude

Latitude and longitude are crucial geographical coordinates used to pinpoint locations on Earth. Latitude indicates how far north or south a place is relative to the equator, while longitude denotes how far east or west a location is from the Prime Meridian.

• Latitude: Measured in degrees, it can range from 0° at the equator to 90° at the poles. Positive values indicate the northern hemisphere, whereas negative values denote the southern hemisphere.
• Longitude: Also measured in degrees, it ranges from 0° at the Prime Meridian to 180° east or west.

In the example of Chicago and Birmingham, we dealt with latitude. Chicago is located at 42° north latitude and Birmingham at 33° north latitude. Given that both cities are directly north-south of one another, only latitude needs to be considered for the distance calculation.

distance calculation

Distance calculation between two points on Earth is an application of understanding spherical geometry. When calculating distance using latitude, we think about how far apart the points are on the curvature of the Earth.
Firstly, we determine the difference in latitude between the locations, like between Chicago (42°) and Birmingham (33°), which is 9°. \Next, to convert this difference into an actual distance, we understand it as a fraction of the Earth's total circular path. Since there are 360° in a full circle, the 9° difference translates to a portion of Earth's circumference. \We perform the following calculation: \\[ \text{Distance} = \left( \frac{9}{360} \right) \times 2\pi \times 3900 \] \This formula calculates the arc distance along the surface of the Earth between the two given latitudes, providing an approximation since the Earth isn’t a perfect sphere. This problem showcases how precalculus connects maths with practical real-world geography.