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Problem 905

# What is the length of the shortest path from the North Pole to the South Pole? (Take the diameter of the earth to be 8,000 miles.)

Expert verified
The shortest path from the North Pole to the South Pole along the Earth's surface is approximately 12,566.37 miles.
See the step by step solution

## Step 1: Understand the problem in spherical geometry context

Since both the North Pole and the South Pole are located on the Earth's surface, we are dealing with a spherical geometry problem. In spherical geometry, the shortest distance between two points is the great circle distance, which is the angular distance along the great circle connecting the two points.

## Step 2: Recall the great circle distance formula

The great circle distance is found by taking the Earth's radius and multiplying it by the central angle, measured in radians, between the two points. The formula for great circle distance is: d = R × θ where d is the great circle distance, R is the Earth's radius (approximately 4,000 miles since the diameter given is 8,000 miles), and θ is the central angle between the North Pole and South Pole in radians.

## Step 3: Calculate the central angle θ between the North Pole and the South Pole

Since the North Pole and South Pole are located 180 degrees apart (as they are on the exact opposite sides of the Earth), the central angle θ will also be 180 degrees. However, we need to convert this to radians to plug it into our great circle distance formula. To convert from degrees to radians, we will use the following equation: radians = (degrees × π) / 180 So, θ in radians is: θ = (180 × π) / 180 θ = π radians

## Step 4: Calculate the great circle distance

Now that we have all the necessary information, we can plug in the values into the great circle distance formula: d = R × θ d = 4,000 miles × π d ≈ 12,566.37 miles Thus, the shortest path from the North Pole to the South Pole along the Earth's surface is approximately 12,566.37 miles.

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