Problem 103 - 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. New York City, New York, 41 \(^{\circ} \mathrm{N}\), and Lima, Peru, \(12^{\circ} \mathrm{S}\)

Step-by-Step Solution

Answer

The distance between New York City and Lima is approximately 5900 km.

Step 1: Identify the Latitude Difference

First, recognize the latitude of New York City is 41°N and the latitude of Lima is 12°S. To find the distance, we need the absolute difference between these latitudes. Calculate this difference: \[ \text{Latitude Difference} = 41 + 12 = 53^{\circ}\] The latitudes are added because they lie in opposite hemispheres.

Step 2: Convert Latitude Difference to Radians

The arclength formula requires angles in radians. Convert the degrees to radians using the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]Substitute 53°:\[ 53 \times \frac{\pi}{180} = \frac{53\pi}{180} \approx 0.925\, \text{radians}\]

Step 3: Calculate the Distance Using Earth’s Radius

To find the surface distance between the two cities along the Earth's surface, use the formula: \[ \text{Distance} = \text{Radius} \times \text{Angle in Radians}\]With a radius of 6400 km, the calculation becomes:\[ \text{Distance} = 6400 \times 0.925 \approx 5920\, \text{km}\]

Step 4: Round to the Nearest Hundred Kilometers

Round the calculated distance to the nearest hundred kilometers. \[ 5920 \approx 5900\, \text{km}\] The distance between New York City and Lima is approximately 5900 km.

Key Concepts

Latitude Conversion to Radians

Latitude is typically measured in degrees, but when calculating distances on the Earth's surface, it's essential to convert these degrees into radians. This is because radian measure is a natural unit of angular measurement used in various mathematical calculations, such as the arclength formula. Converting degrees to radians involves using the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]- For example, if you have a latitude difference of 53 degrees, you substitute this value into the formula, yielding \( 53 \times \frac{\pi}{180} \approx 0.925 \) radians.Converting to radians offers a meaningful way to relate the angle to the circle's radius, crucial for determining the actual distance between two points.

Arclength Formula

The arclength formula is a key mathematical tool for finding distances along the curved surface of a sphere, like Earth. It states that the distance along a path traced by a central angle is the product of the radius and the angle (in radians). The formula is:\[ \text{Distance} = \text{Radius} \times \text{Angle in Radians} \]- Consider the scenario where the radius of the Earth is 6400 km and the latitude conversion resulted in an angle of approximately 0.925 radians.- Applying the formula, the distance is calculated as \(6400 \times 0.925 \approx 5920\) km.By using the arclength formula, you harness the power of basic geometry to make meaningful insights into geographical distances.

Hemispheres and Latitude

Understanding hemispheres and latitude is vital when calculating the distance between two cities because latitude angles measure how far a location is from the Equator. Latitudes north of the Equator are positive, while those south are negative.- Cities in different hemispheres, like New York City (41°N) and Lima (12°S), have their latitudes added to find the total difference.For our example, the total latitude difference is calculated as: \( 41 + 12 = 53^{\circ} \).This approach ensures that the calculation correctly reflects the geographical span between the two distinct hemispheres.