Assume that the Earth is spherical and recall that latitudes range from 0° at the Equator to 90° N at the North Pole. Consider Dubuque, Iowa (42.50° N latitude), and Guatemala City (14.62° N latitude). The two cities lie on approximately the same longitude. Do not neglect the curvature of the Earth in determining the following.
If one could burrow through the Earth and dig a straight-line tunnel from Dubuque to Guatemala City, how long would the tunnel be?
-->I know the answer for above and it is
3072992.8 m..i need help with part b
b)From the point of view of the digger, at what angle below the horizontal would the tunnel be directed?
If the tunnel length is L and the Earth radius is R, the angle below horizontal is
arcsin [L/(2R)]
Prove it to yourself by drawing a circular arc and a straight line between the two cities, with the center of the Earth as the center of the arc. A line from the center of the Earth to the center of the straight line (tunnel) forms a right triangle with hypotenuse R, angle A at the Earth center, and one side equal to L/2.
A = arcsin [L/(2R)]
Look up R. It is about 6.4*10^6 m
I get about 29 degrees. .
for L, are you using the answer from part a?? when i use that i get around 14 degrees
To find the angle that the tunnel would be directed below the horizontal, we can use the concept of the tangent of an angle. Here are the steps to calculate the angle:
1. Determine the distance between the two cities in a straight line. In this case, the distance is already provided as 3072992.8 meters.
2. Assume the Earth is a perfect sphere, and the tunnel is a straight line connecting Dubuque to Guatemala City.
3. Divide the distance by the radius of the Earth to find the angular distance covered. The radius of the Earth is approximately 6371 kilometers, which is equal to 6371000 meters. The angular distance covered can be found using the formula: angular distance = distance / radius. In this case, the angular distance is: 3072992.8 / 6371000 ≈ 0.482 radians.
4. The angle below the horizontal can be found by subtracting the angular distance from a right angle (90 degrees or π/2 radians). Thus, the angle below the horizontal is: 90 - 0.482 ≈ 89.518 degrees.
Therefore, from the perspective of the digger, the tunnel would be directed at an angle of approximately 89.518 degrees below the horizontal.