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 this part and its 3072992.8 m...i need help with the next part
From the point of view of the digger, at what angle below the horizontal would the tunnel be directed?
I thought I answered this already.
sin^-1[L/(2R)] is the answer
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To find the angle at which the tunnel would be directed below the horizontal, we can use trigonometry. Let's start by visualizing the situation.
Imagine a right triangle formed by the tunnel, with the horizontal line representing the Earth's surface, and the vertical line representing the direction downward. We want to find the angle between the horizontal line and the tunnel.
Since we know the length of the tunnel is 3,072,992.8 meters and the radius of the Earth is approximately 6,371 kilometers or 6,371,000 meters, we can subtract the radius from the length to determine the height of the triangle.
Height = Length of tunnel - Radius of the Earth
Height = 3,072,992.8 m - 6,371,000 m
Height = -3,298,007.2 meters
Note that the height is negative because the tunnel points downward below the surface.
Now, we can use trigonometry to find the angle. The tangent function relates the opposite side of a right triangle to its adjacent side. In this case, the opposite side is the height of the tunnel, and the adjacent side is the length of the tunnel.
Tangent(θ) = Opposite/Adjacent
Tangent(θ) = -3,298,007.2 m / 3,072,992.8 m
Taking the inverse tangent (arctan) of both sides, we can find the angle θ:
θ = arctan(-3,298,007.2 m / 3,072,992.8 m)
Using a calculator or computer program, we can find the arctan of the above expression, which is approximately -46.81°.
Therefore, from the digger's perspective, the tunnel would be directed at an angle of approximately -46.81° below the horizontal.