Question

The city of Austin, Texas, and Baton Rouge, Louisiana, are located at 30°N latitude. If the central angle formed from the center of the Earth to each of these cities is 6° and assuming the radius of the Earth to be 3960 miles, what is the distance between Austin and Baton Rouge?

Answer 1

The distance between Austin and Baton Rouge, given a central angle of 6° and assuming the Earth's radius is 3960 miles, is approximately 396 miles.

Step-by-step explanation:

To find the distance between Austin and Baton Rouge using the central angle, we can utilize the formula for arc length on a circle:
`\( \text{Arc Length} = \text{radius} * \text{central angle} \)` . Given the radius of the Earth is 3960 miles and the central angle is 6°, the formula becomes
`\( \text{Arc Length} = 3960 * (6)/(360) \)` . Simplifying this gives us the arc length between the two cities.

Converting the central angle from degrees to radians might also help. To do this, we use the formula
`\( \text{radians} = \frac{\text{degrees} * \pi}{180} \)` . So, for a 6° angle, the conversion to radians would be
`\( (6 * \pi)/(180) \)` . This is crucial because the arc length formula often requires the angle to be in radians.

After calculating the arc length using either method, we find the distance between Austin and Baton Rouge along the Earth's surface. This distance gives an approximation of how far apart the two cities are when measured along the Earth's curved surface, considering the angle from the Earth's center to each city. In this case, it results in a distance of approximately 396 miles. This calculation assumes a perfectly spherical Earth, disregarding any local variations due to terrain or irregularities in the Earth's shape.

Answer 2

The distance between Austin, Texas and Baton Rouge, Louisiana, located at 30°N latitude and separated by a 6° central angle, is approximately 131.9 miles, assuming the Earth's radius is 3960 miles.

Step-by-step explanation:

The city of Austin, Texas, and Baton Rouge, Louisiana, are 30°N latitude apart. With a central angle of 6°, we can calculate the distance between these two cities by using the formula for the arc length on a circle, which is arc length = (central angle/360) × circumference. Given that the circumference of the Earth is 2πr, where r is the radius (3960 miles), we can plug these values into the formula:

arc length = (6° / 360°) × (2π × 3960 miles)

First, calculate the fraction of the circumference that the arc covers:

fraction = 6° / 360° = 1/60

Then, calculate the arc distance:

arc length = (1/60) × (2π × 3960 miles) = 2π × 3960 miles / 60

arc length ≈ 2π × 3960 / 60 ≈ 131.9 miles

Therefore, the distance between Austin and Baton Rouge is approximately 131.9 miles.