nI Exercises 65-68, the latitudes of a pair of cities are given. Assume that one city si directly south of the other and that the earth is a perfect sphere of radius 4000 miles. Use the arc length formula in terms of degrees to find the distance between the two cities. 65. The North Pole: latitude 90° north Springfield, Illinois: latitude 40° north

To find the distance between the North Pole (latitude 90° north) and Springfield, Illinois (latitude 40° north), we can use the arc length formula in terms of degrees.

The arc length formula in terms of degrees is given by:

\text{{arc length}} = \left(\frac{{2\pi\text{{r}}}}{{360}}\right)\theta

where r is the radius of the sphere and θ is the central angle (in degrees).

Given that the radius of the Earth is 4000 miles and the latitudes of the two cities, we can substitute these values into the formula:

For the North Pole:

\text{{arc length}}_{\text{{North Pole}}} = \left(\frac{{2\pi\text{{(4000)}}}}{{360}}\right)(90) \approx 6,283.19 \text{{ miles}}

For Springfield, Illinois:

\text{{\text{arc length}}}_{\text{{\text{Springfield, Illinois}}}}=\left(\frac{{2\pi\text{{(4000)}}}}{{360}}\right)(40)\approx2,792.53\text{{\text{ miles}}}

To find the distance between the two cities, we subtract the arc length of Springfield, Illinois from the arc length of the North Pole:

\text{{Distance between the cities}} = \text{{arc length}}_{\text{{North Pole}}} - \text{{arc length}}_{\text{{Springfield, Illinois}}}

=6,283.19\text{{\text{ miles}}}-2,792.53\text{{\text{ miles}}}

=3490.66\text{{\text{ miles}}}

Answer: The distance between the North Pole and Springfield, Illinois is approximately 3490.66 miles.