Problem 45 - Algebra 2 and Trigonometry

Question

Latitude represents the measure of a central angle with vertex at the center of the earth, its initial side passing through a point on the equator, and its terminal side passing through the given location. (See the figure.) Cities A and \(\mathrm{B}\) are on a north-south line. City \(\mathrm{A}\) is located at \(30^{\circ} \mathrm{N}\) and City \(\mathrm{B}\) is located at \(52^{\circ} \mathrm{N}\) . If the radius of the earth is approximately \(6,400\) kilometers, find \(d\) , the distance between the two cities along the circumference of the earth. Assume that the earth is a perfect sphere.

Step-by-Step Solution

Answer

The distance is approximately 2457.41 km.

Step 1: Understand the Concept of Latitude

Latitude measures the angle formed by a location on Earth relative to the equator. City A is at \(30^{\circ} N\) and City B is at \(52^{\circ} N\). This means we need to find the linear distance along the curve of Earth's surface between these two latitudes.

Step 2: Determine the Angular Separation

To calculate the angular separation between City A and City B, subtract the latitude of City A from the latitude of City B: \(52^{\circ} - 30^{\circ} = 22^{\circ}.\) This gives us the angle through which we have to travel along Earth's circumference.

Step 3: Convert Angular Separation to Radians

Since the distance along a circular path is calculated using radians, convert \(22^{\circ}\) into radians using the formula \(\theta (\text{radians}) = \theta (\text{degrees}) \times \frac{\pi}{180}\). Thus, \(22^{\circ} = 22 \times \frac{\pi}{180} \approx 0.383972\) radians.

Step 4: Apply the Arc Length Formula

The arc length \(d\) can be calculated using the formula \(d = r \cdot \theta\), where \(r\) is the radius of Earth and \(\theta\) is the angle in radians. Substituting the known values, \(d = 6400 \cdot 0.383972 \approx 2457.41\) kilometers.

Step 5: Conclusion

The distance \(d\) between City A and City B along the Earth's surface is approximately \(2457.41\) kilometers.

Key Concepts

Latitude

Latitude is a way of expressing where a place is located on Earth relative to the equator. Think of Earth divided into horizontal rings, with the equator being the widest circle that divides Earth into the Northern and Southern Hemispheres. Latitude is measured as the angle that these rings make with the equator.
- It starts at 0° at the equator and goes up to 90° at the poles (either north or south). - North of the equator is labeled with "N" for north, and south is labeled with "S" for south.
In the exercise, City A is located at 30° N, making it moderately north of the equator, while City B is further north at 52° N. To find out how far apart these two cities are, we need to calculate the difference in their latitudinal angles, referred to as "angular separation." This difference provides us with the portion of the Earth's surface we need to measure.

Radians

Radians are an alternate way to measure angles, besides degrees. While degrees divide a circle into 360 equal parts, radians divide it using the circle's radius. One complete revolution around a circle is equal to 2π radians. This means:
- 180° is equivalent to π radians
- 1 radian is approximately 57.3°
In mathematical calculations, especially those involving circular motion and geometry, radians are preferred due to their natural alignment with the properties of circles.
To convert degrees to radians, use the formula: \[ \theta \,(\text{radians}) = \theta \,(\text{degrees}) \times \frac{\pi}{180}\]In the exercise, we converted the angular separation of 22° into radians, yielding approximately 0.384 radians, to calculate the distance using the arc length formula.

Arc Length

Arc length represents the distance along a curved line or segment of a circle. Imagine if you had a slice of pizza and wanted to measure just the crust, that would be similar to finding the arc length. It is calculated using the formula for a circle's circumference:

• \[\text{Arc Length} = r \cdot \theta\]

where \(r\) is the radius of the circle, and \( \theta \) is the angle in radians.
In the problem, Earth is the circle with a radius of 6400 kilometers. With our calculated angle of 0.384 radians, we plug these values into the arc length formula to find the distance \(d\) between the two cities along Earth's surface. This resulted in an arc length of approximately 2457.41 kilometers.

Central Angle

The central angle is pivotal in determining distances on a circle. This is the angle whose vertex is at the center of the circle, with endpoints on the circle.
In this scenario, the central angle represents the angular distance between the two cities along Earth's surface. It spans from the initial position at City A up to City B, passing through Earth's center.
The exercise uses this angle to gauge how far along the Earth's circumference the two cities are situated from each other. This is why first finding out this angle in degrees and then converting it to radians is essential for accurate distance calculation using the arc length formula, highlighting why understanding the concept of a central angle is important for trigonometry and geographical measurements.