When the time of day for a certain ship at sea is 12 noon, the time of day at the prime meridian (0° longitude) is 5 P.M. What is the ship’s longitude?

The trick with this problem is to figure out the position of the ship in relation to the Prime Meridian, that is, on which side of the Prime Meridian, East or West, you can expect to find the ship.

As you know, longitude expresses the position of a point on Earth's surface in terms of how many degrees East or West relative to the Prime Meridian that point is located.

The Prime Meridian is assigned the value of #0^@# longitude. Now, Earth performs a complete rotation, i.e. #360^@#, in one day, or #24# hours.

This means that you can find Earth's rotation angle per hour by using

#1color(red)(cancel(color(black)("hour"))) * (360^@)/(24color(red)(cancel(color(black)("hours")))) = 15^@"/hour"#

So, the difference between the time at the Prime Meridian, which is given as #"5 pm"#, and the time at the location of the ship, which is given as #"12 pm"#, amounts to a total of #5# hours.

This means that the Earth rotated by a total of

#5color(red)(cancel(color(black)("hours"))) * (15^@)/(1color(red)(cancel(color(black)("hour")))) = 75^@#

But what is the ship's longitude, #75^@# East of the Prime Meridian or #75^@# West of the Prime Meridian?

To figure this out, you can use the fact that the Sun rises in the East and sets in the West, which is equivalent to saying that the Earth rotates from West to East.

Notice that the time in the ship's position is behind the time at the Prime Meridian, i.e. the position of the sun in the sky at the ship's location corresponds to what the position of the sun at the Prime Meridian was five hours earlier.

This means that the Earth will rotate for another five hours until the position of the sun in the sky see by the ship will match that of the sun in the sky at the Prime Meridian five hours earlier.

Since the Earth is rotating from West to East, it follows that the ship must be west of the Prime Meridian, at a longitude of #75^@"W"#.