Problem 101 - 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. Madison, South Dakota, \(44^{\circ} \mathrm{N},\) and Dallas, Texas, \(33^{\circ} \mathrm{N}\)

Step-by-Step Solution

Answer

The distance is approximately 1200 km.

Step 1: Understand the Problem

We need to calculate the north-south distance between two cities: Madison and Dallas, given their latitudes. Assume they are on the same north-south line on the earth's surface. The latitude of Madison is \(44^{\circ} \text{N}\), and that of Dallas is \(33^{\circ} \text{N}\). The Earth's radius is 6400 kilometers.

Step 2: Calculate the Difference in Latitudes

Find the difference between the latitudes of Madison and Dallas. \(44^{\circ} - 33^{\circ} = 11^{\circ}\). This represents the angular difference between the two cities.

Step 3: Convert the Angular Distance to Radians

Since the formula for arc length requires radians, convert the angular distance from degrees to radians. Use the conversion: \[1^{\circ} = \frac{\pi}{180} \text{ radians}\]Thus, \[11^{\circ} = 11 \times \frac{\pi}{180} \approx 0.192 \, \text{radians}\]

Step 4: Use the Arc Length Formula

To find the distance between the cities along the surface of the Earth (an arc), use the formula for arc length:\[\text{Arc Length} = \text{radius} \times \text{angle in radians}\]Here, the radius is 6400 km and the angle is 0.192 radians. Thus,\[\text{Arc Length} = 6400 \times 0.192 \approx 1228.8 \, \text{km}\]

Step 5: Round the Distance

Round the calculated distance to the nearest hundred kilometers:\[1228.8 \text{ km} \approx 1200 \text{ km}\]

Key Concepts

Arc Length

Understanding arc length is essential when you are dealing with distances on a curved surface, like the Earth. Imagine the Earth as a circle; any distance along its surface can be visualized as part of a circle's circumference. This fragment of a circle is called an 'arc'.
To calculate the arc length, you need two values: the radius of the circle and the angle in radians subtended by the arc at the circle's center. The formula used is \( \text{Arc Length} = \text{radius} \times \text{angle in radians} \). For this exercise, we have Earth's radius as 6400 km.
Key steps in the calculation:

• First, find the angular distance between the two points (cities), which in this case is provided by the difference in latitude.
• Convert this angular measurement from degrees into radians, since the formula requires radians.
• Multiply the radius by the angle in radians to get the arc length, which gives the distance along the Earth's surface.

In summary, arc length allows us to determine the surface distance between two points on a spherical object like Earth.

Latitude and Longitude

Latitude and Longitude are a system used to locate any point on the Earth's surface. They form a grid-like system with lines that run both horizontally and vertically. In this context, we focus on latitude, which indicates how north or south a location is relative to the equator.
Latitude lines are imaginary circles that run parallel to the equator, with the equator itself being at 0° latitude. As you move towards the north or south poles, these values increase to 90° North or 90° South.
In the exercise, we distinguish between two locations by their latitudes: Madison at 44°N and Dallas at 33°N. The movement north or south along these lines reflects a change in latitude, which is crucial when calculating physical distances between different points on Earth using the arc length method.

Unit Conversion

Unit Conversion is a fundamental mathematical skill, especially when working with measurements. It becomes pivotal when you need calculations with specific units like degrees and radians.
In many calculations involving angles, converting degrees to radians is necessary since many trigonometric formulas, including those for arc length, require angles in radians. The conversion factor from degrees to radians is \( \frac{\pi}{180} \). For example, converting 11° to radians involves multiplying by this factor, resulting in approximately 0.192 radians.
Steps for conversion:

• Identify the original unit (in this case, degrees).
• Apply the conversion factor to switch from degrees to radians.
• Proceed with using the converted measure in subsequent calculations.

Understanding and performing unit conversions accurately ensures that calculations are consistent and reliable, allowing for proper application of formulas across various scenarios.