Two warships, A and B, set sail from the geographical coordinates (Lat=36.81631,Lon=30.70623) and (Lat=36.60179,Lon=36.07906) , respectively. They need to meet at the geographical location (Lat=35.38189,Lon=33.33106) ) after 4 hours. What should be the departure bearing (Azimuth Angles) and the minimum sailing speeds of warships A and B, assuming the Earth's radius is 6371 km?

To find the departure bearing and the minimum sailing speeds of warships A and B, we can use the great-circle distance formula and the spherical law of cosines.<br /><br />Let's denote the coordinates of warships A and B as $(Lat_A, Lon_A)$ and $(Lat_B, Lon_B)$, respectively. The coordinates of the meeting point are $(Lat_M, Lon_M)$.<br /><br />The great-circle distance formula is given by:<br /><br />$$d = R \cdot \arccos(\sin(Lat_A) \cdot \sin(Lat_M) + \cos(Lat_A) \cdot \cos(Lat_M) \cdot \cos(Lon_A - Lon_M))$$<br /><br />where $R$ is the Earth's radius.<br /><br />Using the spherical law of cosines, we can find the minimum sailing speeds of warships A and B:<br /><br />$$\cos(d) = \cos(d_A) \cdot \cos(d_B) + \sin(d_A) \cdot \sin(d_B) \cdot \cos(\theta)$$<br /><br />where $d_A$ and $d_B$ are the distances traveled by warships A and B, respectively, and $\theta$ is the angle between their paths.<br /><br />Since the warships meet after 4 hours, we have $d_A = v_A \cdot t$ and $d_B = v_B \cdot t$, where $v_A$ and $v_B$ are the speeds of warships A and B, respectively, and $t$ is the time taken to meet.<br /><br />Substituting the get:<br /><br />$$\cos(d) = \cos(v_A \cdot t) \cdot \cos(v_B \cdot t) + \sin(v_A \cdot t) \cdot \sin(v_B \cdot t) \cdot \cos(\theta)$$<br /><br />Solving this equation for $v_A$ and $v_B$, we can find the minimum sailing speeds of the warships.<br /><br />The departure bearing (azimuth angle) can be found using the following formula:<br /><br />$$\tan(\theta) = \frac{\sin(Lat_M - Lat_A)}{\cos(Lat_M) \cdot \cos(Lon_M - Lon_A) - \sin(Lat_M) \cdot \sin(Lat_A)}$$<br /><br />where $\theta$ is the angle between the paths of the two warships.<br /><br />Using this formula, we can find the departure bearing for warships A and B.