Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the great-circle distance from St. Paul (longitude \(93.1^{\circ} \mathrm{W}\), latitude \(\left.45^{\circ} \mathrm{N}\right)\) to Turin, Italy (longitude \(7.4^{\circ} \mathrm{E}\), latitude \(\left.45^{\circ} \mathrm{N}\right) .\)

Step-by-Step Solution

Answer

The great-circle distance from St. Paul to Turin is approximately 7323 km.

Step 1: Convert Degrees to Radians

To use the formulas effectively, we need to convert the geographic coordinates from degrees to radians. Recall that 1 degree is equal to \( \frac{\pi}{180} \) radians. For St. Paul, we convert:\[l_1 = 93.1 \cdot \frac{\pi}{180} \approx 1.625 \, \text{radians (W is negative)}\]For Turin:\[l_2 = 7.4 \cdot \frac{\pi}{180} \approx 0.129 \, \text{radians (E is positive)}\]For both locations, the latitude \( \phi = 45^{\circ} \) converts to:\[\phi = 45 \cdot \frac{\pi}{180} \approx 0.785 \, \text{radians}\]

Step 2: Apply the Great-Circle Distance Formula

The great-circle distance formula is given by:\[\Delta\lambda = l_2 - l_1\]\[\text{distance} = R \cdot \arccos(\sin(\phi_1) \cdot \sin(\phi_2) + \cos(\phi_1) \cdot \cos(\phi_2) \cdot \cos(\Delta\lambda))\]where \( R \approx 6371 \, km \) is the Earth's radius and \( \Delta\lambda = 0.129 + 1.625 \approx 1.754 \, radians \).

Step 3: Calculate the Great-Circle Distance

Plug in the values from the previous step into the formula:\[\text{distance} = 6371 \cdot \arccos(\sin(0.785) \cdot \sin(0.785) + \cos(0.785) \cdot \cos(0.785) \cdot \cos(1.754))\]Calculate approximately:\[= 6371 \cdot \arccos(0.5 + 0.5 \cdot (-0.184))\approx 6371 \cdot \arccos(0.408)\approx 6371 \cdot 1.15 \, \approx 7323 \, km\]

Step 4: Verify Calculations

Re-check each step, especially the conversion of degrees to radians and the application of trigonometric identities in the formula. Ensure that the inverse cosine function has been applied correctly for obtaining an angle in radians.

Key Concepts

Latitude and Longitude Conversion

Latitude and longitude are essential in identifying locations on Earth's surface. They are expressed in degrees, where latitude ranges from 0° at the Equator to 90° at the poles, and longitude extends from 0° at the Prime Meridian to 180° east or west.
For calculations like the great-circle distance, coordinates are often converted from degrees to radians. This is because mathematical operations, particularly trigonometric functions, are more straightforward in radians. To convert degrees to radians, you multiply the degree value by \( \frac{\pi}{180} \).

• Latitude Example: For 45° N, the formula is \( 45 \cdot \frac{\pi}{180} = 0.785 \, ext{radians} \).
  
• Longitude Example: St. Paul's 93.1° W becomes \( 93.1 \cdot \frac{\pi}{180} \approx 1.625 \, ext{radians} \) (negative for the west), and Turin's 7.4° E is \( 7.4 \cdot \frac{\pi}{180} \approx 0.129 \, ext{radians} \) (positive for the east).

Converting these angles accurately is crucial for precise calculations when finding distances between two points on Earth.

Radian Measurement

Radians provide a different way of expressing angles apart from degrees. In contrast to degrees, which circle around 360°, radians simplify the calculations by considering the arc length equal to the radius.
For context, one full circle in radian measure is \( 2\pi \) radians because the circumference is \( 2\pi imes ext{radius} \). Since 360° represents a full circle, the number of radians in a circle is also 360°/degree to radians conversion, therefore, \( \pi \text{ radians} = 180° \).

• Reason to Use Radians: Most trigonometric calculations are simpler in mathematical formulas when using radian measure. This is because the radian is a unit that naturally fits into the trigonometric functions, aligning with the circle's geometry.

The use of radians is especially advantageous in calculus and physics, where angles and angular speeds frequently appear in wave equations and circular motion.

Trigonometry in Geography

Trigonometry plays a crucial role in geography, particularly in calculating distances on a globe like Earth, which can't be accurately represented on a flat map. The concept of a great-circle distance uses trigonometric principles to find the shortest path between two points on a sphere.
In the formula for great-circle distance, we utilize the cosine and sine functions to factor in the Earth's curvature. Sine and cosine are fundamental trigonometric functions that relate to the sides and angles of triangles. Here's how they apply in our context:

• Angle Differences: The formula involves calculating the cosine of the difference in longitude, \( \Delta\lambda\), affecting the computed distance.
• Cosine Function: We use \( \cos(\phi_1) \cdot \cos(\phi_2) \) to adjust for the latitude's influence on the path's angle.

The great-circle formula, thus combining sine and cosine, accounts for the earth's spherical shape, ensuring more accurate geographical distance calculations than straight-line distances used for flat maps.

Earth's Radius

The Earth's radius is a fundamental component when calculating distances on the planet. It's understood that Earth isn't a perfect sphere but a slightly oblate spheroid; however, for many calculations, a simplified average radius of 6371 km is used.
This radius represents the mean distance from the Earth's center to its surface, simplifying calculations without significantly affecting accuracy for general use in navigation and geolocation.

• Usage in Great-Circle Formula: In the great-circle distance formula, the Earth's radius (R) is multiplied by the arccosine term, which calculates the angle in radians along the circle compared to the total 360 degrees around the Earth.
• Implications: Understanding Earth's radius helps in comprehending how distances and speeds over the Earth's surface translate into actual travel times and navigation routes.

Knowing the Earth's average radius is vital for calculations involving large-scale distances, such as air travel routes, satellite positioning, and telecommunications.