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?

42.50 - 14.62 = 27.88 °

The circumference of a circle = 2 r pi

The formula for the arc length of a circle:

L = 2 r pi * theta / 360 ° = r pi * theta / 180 °

The average radius of the Earth is 6371 km

L = 6371 * pi * 27.88 ° / 180 ° = 3100.115 km

Approx. 3100 km

L = The length of that arc between Dubuque and Guatemala City

The length of the tunnel = The chord length :

c = 2 r sin ( theta / 2 )

c = 2 * 6371 * sin ( 27.88 ° / 2 )

c = 2 * 6371 * sin 13.94 °

c = 2 * 6371 * 0.24091

c = 3069.67522 km

Approx. 3070 km

To determine the length of a straight-line tunnel from Dubuque to Guatemala City, we need to calculate the distance along the curve of the Earth's surface between the two cities. Here's how you can do it:

1. Calculate the circumference of the Earth: The Earth's circumference is approximately 40,075 kilometers (or 24,901 miles). This value represents the distance around the Earth at the Equator.

2. Calculate the difference in latitudes between Dubuque and Guatemala City: Subtract the latitude of Dubuque (42.50° N) from the latitude of Guatemala City (14.62° N). This gives us a latitude difference of 27.88°.

3. Determine the fraction of the Earth's circumference covered by the latitude difference: Divide the latitude difference by 360° since the Earth's circumference represents a full circle (360°). In this case, the fraction would be 27.88° / 360°, which is approximately 0.0774.

4. Calculate the length of the tunnel: Multiply the fraction calculated in step 3 by the circumference of the Earth. This gives us the length of the tunnel in kilometers (or miles). So, tunnel length = 0.0774 * 40,075 km = approximately 3,100 km.

Therefore, if one could burrow through the Earth from Dubuque to Guatemala City, the tunnel would be approximately 3,100 kilometers long.