Homework Help: Math

Posted by rachel on Saturday, March 3, 2012 at 8:31pm.

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?

Math - Anonymous, Saturday, March 3, 2012 at 10:28pm
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
Math - Anonymous, Saturday, March 3, 2012 at 10:52pm
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

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