Section   16.2Latitude and Longitude

1. (Exercises 3 and 4) If two cities lie on the same meridian, what will be the distance between them if they are located at the following latitudes and longitudes: A is at (82°S, 87°W) and B is at (38°S, 87°W).

Since the longitude is the same for each location, we will be traveling along a great circle where 1 degree is equal to 60 nautical miles. The distance traveled would then be the number of degrees difference in latitude times 60 nautical miles/degree.

distance = (82°W - 38°W) (60 nautical miles/degree)

= (44 degrees) (60 nautical miles/degree) = 2640 nautical miles

Traveling from A to B, we would be traveling south.

This can be converted to land or statute miles by using the ratio:

60 nautical miles = 69 statute miles

distance = (2640 nautical miles)(69 stat.miles / 60 naut.miles) = 3040 statute miles

2. (Exercises 7 and 8) Determine the longitude and latitude of a point on Earth that is directly opposite the Hawaiian Islands, if their location is given as (21°N, 158°W).

First, if you were on opposite sides of Earth at the equator, your latitude would be 0° in both places; likewise, if you were at the North Pole (90°N), the opposite location would be the South Pole (90°S). Opposite latitudes have the same number of degrees but move from one hemisphere to the other, so a point opposite 21°N would be 21°S.

To find the opposite longitude we must go 180° around Earth, so traveling from 158°W we would eventually reach 158° - 180° = - 22°. The negative here means we are on the other side of the prime meridian (0°) by 22°, so we have moved from 158°W to 22°E. The location of the point opposite the Hawaiian Islands is then at (21°S, 22°E).

Section   16.3Time

3. (Exercises 13 and 14) A man on a business trip calls his wife in Ohio at 6:00 P.M. MST from Salt Lake City. What time does she get the call if she is in the Eastern Standard Time zone?

There is 2 hours difference between MST (centered at 105°) and EST (centered at 75°) because the time changes by 1 hour for each 15° that we travel east or west. Since the wife is east of her husband, the time there is later than it is for him. Two hours later means that her call will come at 6:00 + 2 h = 8:00 PM EST.

4. (Exercises 17 and 18) What will be the altitude of the Sun at noon for an observer in Hawaii (21°N, 158°W) on September 21 and on December 21?

On September 21 the declination of the Sun is 0° (it is above the equator), so the angle is easy to calculate. The zenith angle (ZA) for the Sun on this date will be 21° - 0° = 21°, and the altitude will be 90° - ZA = 90° - 21° = 69° This means the noontime Sun will be quite high in the sky, about 69° above the southern horizon.

On December 21 the declination of the Sun is 23.5° This means that the zenith angle will be 21° - 23.5° Since these are on opposite sides of the equator, the zenith angle is really 21° - ( - 23.5°), or 44.5° This makes the altitude 90° - 44.5°, or 45.5°, which places the noontime Sun much lower in the sky, only 45.5° above the southern horizon.

If you have difficulty visualizing this angle, you can draw a circle diagram of Earth, as shown in Figure 16.16 in the textbook, to indicate the various angles involved.

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