Question From Readers # 2: Parallel Sailing: Calculate Latitude

Parallel Sailing Formula

 A ship "A" is on the equator steering 090°T at 16 knots ; while a ship"B" is on the parallel of north latitude, steering 270°T at 12 knots. When "A" makes Dlo of 1', "B" makes a Dlo of 48'. Calculate the latitude of "B".

Given:

Ship A
               Course = 090°
                Speed = 16 knots

Ship B

               Course = 270°
                Speed = 12 knots

What is asked?

Latitude of ship B

Analysis:

The relationship between ships is that when "A" makes Dlo of 1', "B" makes a Dlo of 48'.

This is not quite an easy problem compared to other parallel sailing questions. However, if we could solve for a distance traveled by ship A when she makes a Dlo of 1 nautical mile at the equator, then that will be our key to solving for the latitude of ship B.

This is how...

Solution:

1. Solving for a distance traveled by ship A at the equator when she makes a dlo of 1'.

            Distance = Dlo x Cosine Latitude
                          = 1 nm x Cos 0°               (equator is zero latitude)
                          = 1 nm x 1
            Distance = 1 nm

2. Second, we have to get how much time when ship A travel for 1 nm with the speed of 16 kts.

   Steaming Time = 1 nm               (1 knot is equal to 1 nm/hr)
                             16 nm/hr
                         = 1 nm        
                            16 nm/hr   
                         = 0.625 hr       (convert it to minutes)
                         = 0.625 hr x 60 min = 3.75 mins = 3.75 mins
                                              1 hr       1
                         = 3.75 mins

   So, for ship A with the speed of 16 knots, she could travel 1 nautical mile at the equator for 3.75 mins and makes a  Dlo of 1'. How about ship B?

3. Now for the distance traveled for ship B, we have to use the same steaming time 3.75 mins because of this relationship that when "A" makes Dlo of 1', "B" makes a Dlo of 48'. In parallel sailing, distance is equal to the departure.

           Distance = Speed x Time
                         = 12 x 3.75
                         = 45            is the departure of ship B. Dep = Dist
     
                                               Distance = Dlo x Cos Lat
                                               Depature = Dlo x Cos Lat

4. And now for solving for the latitude of ship B, we have to divide the departure by Dlo, and then inverse cosine the answer.

                Dep = Dlo x Cos Lat

 Dlo x Cos Lat = Dep

 Dlo x Cos Lat = Dep 
        Dlo             Dlo

 Dlo x Cos Lat = Dep 
        Dlo             Dlo

         Cos Lat = Dep    
                        Dlo

         Cos Lat = 45
                        48
                     = 0.9375
                Lat = 0.9375 inv Cos
                Lat = 20° 22' N   is the latitude of ship B

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